1

Asymptotic methods for weakly nonlinear and other water waves

A lecture course given at the International Mathematical Summer Centre (CIME Foundation)

Cetraro, June 2013

by

Prof. Robin Johnson (School of Mathematics & Statistics, Newcastle University, UK)

Outline

We first introduce the model and equations that describe the classical water-wave problem, and then list some of the problems that can be addressed. The particular problems of interest here will be outlined, and the associated non-dimensionalisation and scalings developed. The resulting form of the equations will then be interpreted, and two classical problems mentioned: fully nonlinear long waves and the solitary wave.

As an introduction to the main discussion in these lectures, we first describe, in general terms, the ideas and techniques of asymptotic (parameter) expansions. We then use this approach to develop some important, approximate equations that are generated by the water-wave problem: Korteweg-de Vries (KdV), 2DKdV (or Kadomtsev-Petviashvili (KP)), concentric KdV (cKdV), a nearly-concentric KdV (ncKdV), Boussinesq, 2D Boussinesq and, finally, Camassa-Holm (CH). (The 2DKdV, 2D Boussinesq and ncKdV equations will be presented only in outline in these lectures.) Many of these equations are of completely integrable type i.e. they are ‘soliton’ equations, for which an inverse scattering transform (IST) exists. These important ideas will be briefly described, by presenting some results for our standard soliton equations. (This is a very large and deep topic, the development of which is outside the scope of these lectures, so the background and technical details will be omitted.) We will then briefly describe the problem of modulated waves as they appear in water-wave propagation; this leads to the Nonlinear Schrödinger (NLS) equation, and a two-dimensional variant of this: the Davey-Stewartson (DS) equation. The IST theory for the NLS equation will be mentioned. With this basic material as a background, we then describe how these models can be improved in order to represent, more accurately, physically realistic flows (but always in the absence of viscosity and turbulence!). To this end, we outline the problem of variable depth (associated with the KdV equation), and show that this results in a quite complicated technical problem (which we will not pursue here).

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A more accessible class of problems, of considerable practical and current mathematical interest, involves the inclusion of vorticity (which can be thought of as the prescription of some background flow on which a water wave propagates). In particular, we will present the corresponding problems associated with the KdV, CH, NLS and Boussinesq equations (although that for the NLS will be only briefly mentioned). We conclude this type of problem by discussing the propagation of ring waves on a flow with some prescribe flow (i.e. non-zero vorticity) in a given direction. To complete this introduction to the way in which we can use (formal) asymptotic expansions to extract some details of the flow, we look at two rather different – and rather special – wave-propagation problems: periodic waves with vorticity (which has been of considerable interest, over the last decade or so, from the rigorous viewpoint) and an approach to the problem of edge waves. Much of this material appears in various papers and texts; these lecture notes will contain an extensive list of references for the interested reader. In addition, a few exercises are included (for those so inclined); these are mentioned at the appropriate stage in the text and can be found at the end, and before the list of references.

Lecture 1

1 Governing equations The problem, typically referred to as the classical problem of water waves, takes as its model an incompressible, inviscid fluid with zero surface tension. Further, the water moves over an impermeable bed (which we take here to be stationary), with a constant pressure – atmospheric pressure – at the free surface. The fundamental governing equations are then Euler’s equation and the equation of mass conservation:

D 1

Dp

t ρ= − ∇ +u

F ; 0∇ ⋅ =u (1.1)

with constantap p= = & D

D

hw

t= on ( , )z h t⊥= x (1.2)

and ( )w b⊥ ⊥= ⋅∇u on ( )z b ⊥= x , (1.3)

where (0,0, )g= −F , for constant g, and ρ is the constant density of water; D Dt is the

familiar material derivative. The velocity in the fluid has been written as ( , )w⊥=u u ,

with ( , )h h t⊥= x (the free surface), where ⊥x is the 2-vector perpendicular to the z-

coordinate, with the associated velocity vector ⊥u ; w is the component of the velocity

in the z-direction. It is useful, at this early stage, to retain the two-dimensionality of the surface (even though much of our analysis will be for one-dimensional plane waves) because we need this extra freedom in a couple of calculations that we present here. We shall seek a solution, constructed from a suitable function-set, in the domain

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[ ( ), ( , )]z b h t⊥ ⊥∈ x x and 0t ≥ ,

with, for example, x−∞ < < ∞ (and no y-dependence) or y−∞ < < ∞ and

( , )r y t x≤ < ∞ (e.g. for edge waves propagating in the y-direction). The domain in z describes a region whose upper boundary is the free surface, which we do not know a priori . This exemplifies one of the essential difficulties of the water-wave problem: the determination of the domain where the solution exists is part of the solution process. Although it would be natural to prescribe some general initial state, and then examine how this evolves in time (a very difficult problem), we take a different route here. Our main interest is in the form of governing equation that describes a particular type of water wave. Thus our approach is to emphasise this equation, and how we derive it (and, of course, the assumptions underpinning it); then we may address the question of what type of initial data is required to generate the solutions of the equation. The fundamental assumptions that we have described above reflect the physical reality of these types of fluid flow. The incompressibility of water is well known (Lighthill, 1978): e.g. at 150 atmospheres, water is compressed by considerably less than 1%. The effects of laminar viscosity are important only on length scales much greater than the wave lengths and propagation distances that we envisage; da Silva & Peregrine (1988). Further, for ocean waves that are not close to breaking, the effects of turbulent-mixing (and associated viscosity) are similarly unimportant (Barnett & Kenyon, 1975); thus we may reasonably ignore the rôle of viscosity altogether in our model. We also exclude from our model any wind action on the waves, either by the direct transfer of momentum by pressure changes, or by the shearing action of the air. On the other hand, surface tension is certainly important in the class of waves normally described as ‘capillary’, but these waves have wave lengths of only a few centimetres – and our waves (in the ocean, for example) are typically of wave lengths 100-150m; Barnett & Kenyon (1975), Lighthill (1978), Johnson (1997). Our focus of attention will be on the generation and propagation of gravity waves, which arise by virtue of the balance between gravity and the inertia of the system, and so capillary waves will also be absent from our model. At this point, we can begin to appreciate the various types of problem that could now be examined, each being some appropriate solution of this system but satisfying suitable boundary and initial conditions. Some of the classical problems that we will not address in this course include: ship-wave pattern, sloshing in tanks, hydraulic jump and bore, short-crested waves, tsunamis, storm surges, Kelvin-Helmholtz instability, Rayleigh-Taylor instability, propagation of capillary waves, general energy and momentum flux in wave propagation and, of course, the effects of viscosity on suitably long scales. Far more background to the theory of many types of water-wave propagation can be found in: Stoker (1957), Crapper (1984), Dean & Dalrymple (1984), Mei (1989), Debnath (1994), Johnson (1997). The next stage in any problem of this type – or, indeed, in a systematic approach to any problem in applied mathematics – is to non-dimensionalise the set (1.1)-(1.3) by introducing suitable general scales that describe the class of problems under consideration.

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Fig. 1 The coordinates and typical scales

Let 0h be the depth of the water in the absence of waves for some fixed (perhaps

average) level bottom, and λ an average or typical wavelength of the wave (see Figure

1); an associated speed scale is then 0gh , with a corresponding time scale 0ghλ .

This speed scale, however, is used only for the horizontal velocity components ( ( , )u v⊥ =u ); in order to be consistent with the equation of mass conservation (and so,

equivalently, consistent with the existence of a stream function), the vertical

component of the velocity (w) is non-dimensionalised by using 0 0h gh λ . Thus we

non-dimensionalise according to the transformation

λ⊥ ⊥→x x , 0z h z→ , ( )0t gh tλ→ ,

0gh⊥ ⊥→u u , ( )0 0w h gh wλ→ , 0b h b→ , (1.4)

where ‘→ ’ is to be read as ‘replace by’ (so that, for convenience, the current notation is retained, but now all the variables are non-dimensional versions of those introduced earlier). Further, we also introduce 0h h aη= + and 0 0( )ap p g h z gh pρ ρ→ + − + , (1.5)

where a is a measure of the wave amplitude and the new p is a non-dimensional pressure that measures the deviation away from the hydrostatic pressure distribution. (Note that, in the transformation for p, the original, dimensional z is used.) We now use (1.4, 1.5) in equations (1.1)-(1.3) to give

D

Dp

t⊥

⊥= −∇u, 2 D

D

w p

t zδ ∂= −

∂, 0∇ ⋅ =u , (1.6)

with

p εη= & ( )wt

ηε η⊥ ⊥∂ = + ⋅∇ ∂

u on 1 ( , )z tεη ⊥= + x (1.7)

and ( )w b⊥ ⊥= ⋅∇u on .z b= (1.8)

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Here, we have introduced the two familiar, and fundamental, parameters that characterise the classical water-wave problem: 0a hε = , the amplitude parameter, and

0hδ λ= , the long wavelength (or shallowness) parameter. The final stage in the

construction of an appropriate form of these equations is to scale with respect to ε. The case 0ε = recovers undisturbed conditions in the absence of any waves; indeed, as 0ε → , so the disturbance associated with the propagating wave vanishes. Although there are many problems for which we may not wish to take this limit – we could elect to examine the ‘fully nonlinear’ problem – the equations must be consistent with this choice. Thus we further redefine our variables according to the additional transformation ( , , ) ( , , )w p w pε⊥ ⊥→u u (1.9)

when the underlying flow configuration is that of stationary water; we will allow for an existing background vorticity later. The final form of our governing equations, at this stage, therefore becomes

D

Dp

t⊥

⊥= −∇u, 2 D

D

w p

t zδ ∂= −

∂, 0∇ ⋅ =u , (1.10)

with

p η= & ( )wt

η ε η⊥ ⊥∂= + ⋅∇∂

u on 1z εη= + (1.11)

and ( )w b⊥ ⊥= ⋅∇u on z b= , (1.12)

where ( )D

Dt tε∂≡ + ⋅∇

∂u .

This version of the problem allows for the identification of various problems of practical interest, accessed by making suitable choices of the two parameters: neglecting either one or the other, for example, or choosing some special relation between them. Thus, to be specific, we have two standard approximations to these equations: (a) 0ε → (δ fixed): the linearised problem; (b) 0δ → (ε fixed): the long-wave or shallow-water problem. The first problem clearly recovers the most general linear problem, whereas the second is fully nonlinear, but the pressure correction (due to the passage of the wave) is missing: there is no dispersion in this case. We see how these simple approximations can be used, and expanded, in some of what we do later. In the case that the flow is strictly irrotational (so = ∇× =ω u 0 i.e. φ= ∇u ), then an alternative formulation of this problem is available (and sometimes this is more useful than the version above, which is based on the Euler equation). However, much of the work here will be developed from the Euler equation, even if the flow is irrotational, so that direct comparison with flows with vorticity can readily be made. For irrotational flow, we have

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2 2 0zzφ δ φ⊥+ ∇ = (1.13)

with ( ){ }2z tφ δ η ε η⊥ ⊥= + ⋅∇u & 2 2

2

1 1( ) 0

2t zφ η ε φ φδ⊥

+ + ∇ + =

on 1z εη= +

(1.14)

and 2( )z bφ δ ⊥ ⊥= ⋅∇u on z b= . (1.15)

These equations have been non-dimensionalised and scaled exactly in accordance with equations (1.10)-(1.12); the second equation in (1.14) is the pressure equation (sometimes referred to as the ‘unsteady Bernoulli’ equation) evaluated at the free surface, with constant atmospheric pressure imposed. In these equations, we have introduced subscripts to denote partial derivatives. For some applications, it is relevant (and useful) to remove the explicit dependence on δ; this is accomplished by performing one more transformation:

δε⊥ ⊥=x X , t T

δε

= , w Wε

δ= (1.16)

which produces the set (1.10)-(1.12) with 2δ replaced by ε, for arbitrary δ. Note that,

because the terms in 2δ (now ε) are associated with the dispersive property of the wave, 0ε → will describe weakly nonlinear, weakly dispersive waves. (This choice of variables is equivalent to using the single scale length 0h , rather than both0h and λ.) In

this case, we consider only 0ε → , for any δ, and so we can no longer access the problem of 0δ → (ε fixed) for which we must revert to the earlier version of the governing equations. Nevertheless, this alternative version of the equations is useful in a description of, for example, the classical derivation of the Korteweg-de Vries equation; see Lecture 2. This, for many examples, will be coupled with the restriction to one-dimensional (plane wave) propagation.

2 Two classical problems in the theory of water waves There are many problems, of both practical and mathematical interest, that are generated by our governing equations (even though we have chosen to use the simplest model for the fluid and its surface boundary conditions). Although the main thrust in this course is to present the techniques and results associated with special (e.g. ‘soliton’) equations, we start with two different types of problem that, perhaps, may exemplify the broad sweep of the subject. 2.1 Nonlinear, long waves This problem is based on the long-wave ( 0δ → ) version of the equations, for waves propagating in only one dimension; thus equations (1.10)-(1.12) become ( )t x z xu uu wu pε+ + = − ; 0zp = ; 0x zu w+ = , (2.1)

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with p η= & t xw uη ε η= + on 1z εη= + (2.2)

and 0w = on 0z = . (2.3) Here, we have written ( ,0)u⊥ =u , suppressed any dependence on y (so the wave is

propagating in the x-direction: ( , )x tη η= ) and we have taken the bottom to be simply 0z = (i.e. 0b ≡ ).

The essential character of the long wave is now evident: the second equation in (2.1) gives, directly, that ( , )p p x t= and then the surface boundary condition, (2.2), requires

( , )p x tη= ( 0 1z εη≤ ≤ + ). Thus the pressure in the fluid is provided solely by the hydrostatic pressure distribution i.e. the pressure below the surface depends only on the distance below the surface. The equations then reduce to ( )t x z xu uu wuε η+ + = − ; 0x zu w+ = , (2.4)

with t xw uη ε η= + on 1z εη= + and 0w = on 0z = . (2.5)

Now these equations admit a solution for which ( , )u u x t= (which is the only solution if, somewhere, u is independent of z, for then it will remain so); this is simply stating that the flow is irrotational, which is the additional simplification that we invoke. (The vorticity is – here in the y-direction – equal to (z xu w− ), which is zero if, somewhere,

there is the uniform flow ( , )u u x t= and w = 0.) Thus, from (2.4), we see that

( )z xw u= − is not a function of z, so the solution for w which satisfies the boundary

conditions is

1t xu

w zη ε η

εη+ = +

;

the two equations in (2.4) therefore become

0t x xu uuε η+ + = ; (1 ) 0x t xu uεη η ε η+ + + = .

There is no assumption here about the size of ε: we have retained ‘full’ nonlinearity, so we could set 1ε = (and then we are using simply 0h as the length scale i.e. no ‘a’);

indeed, it is convenient to write the surface as 1 ( , ) ( , )x t h x tεη+ = to obtain 0t x xu uu h+ + = ; ( ) 0t xh uh+ = . (2.6)

This pair of equations, (2.6), can be solved by the method of characteristics; to

accomplish this, we first introduce ( , )c x t h= (and we certainly have 0h ≥ , because this is the total depth of the water). Then c is the speed of propagation (in non-dimensional variables) of infinitesimal waves in water of depth h. The two equations may then be written

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2 0t x xu uu cc+ + = and 22 2 0t x xcc c u ucc+ + = or (2 ) (2 ) 0t x xc u c cu+ + = ; (2.7)

these are added to give ( 2 ) ( 2 ) 2 0t x x xu c u u c cc cu+ + + + + = ,

and subtracted to give ( 2 ) ( 2 ) 2 0t x x xu c u u c cc cu− + − + − = .

These, in turn, can be written as

( ) ( 2 ) 0u c u ct x

∂ ∂ + + + = ∂ ∂ and ( ) ( 2 ) 0u c u c

t x

∂ ∂ + − − = ∂ ∂ ,

and so 2 constantu c+ = on lines d

d

xu c

t= + , (2.8)

and 2 constantu c− = on lines d

d

xu c

t= − . (2.9)

This provides the basis for the construction of the solution using the characteristic lines and Riemann invariants; see Stoker (1957), Courant & Friedrichs (1967), Garabedian (1964). [Exercise 1.] 2.2 The solitary wave The solitary wave, as it appears in water waves, has a long and illustrious history, going back to J. Scott Russell’s report (1844); see, for example, Johnson (1997). We will be presenting some aspects of the solitary wave of small amplitude later (because it plays an important rôle in soliton theory and the KdV equation). Here, we make a few observations about some more general properties of this wave, not restricted to small amplitude. This wave, in its simplest manifestation, propagates in irrotational flow (and we will consider the case of the wave propagating into stationary water), so we will start with equations (1.13)-(1.15):

2 2 0zzφ δ φ⊥+ ∇ =

with ( ){ }2z tφ δ η ε η⊥ ⊥= + ⋅∇u & 2 2

2

1 1( ) 0

2t zφ η ε φ φδ⊥

+ + ∇ + =

on 1z εη= +

and 0zφ = on 0z = ,

where 0b ≡ . This set admits solutions that represent (one-dimensional) waves of elevation that propagate with unchanging form (shape) – travelling-wave solutions – which decay exponentially away from the peak. [Exercises 2 and 3.] There is also a limit on the amplitude of the wave; see Stokes (1880), Varvaruca (2009), Varvaruca & Weiss (2011). The small-amplitude approximation is accessed by imposing 0ε → , but we will not restrict ourselves to this here, so we set 1ε = ; δ will also be arbitrary. A number of general properties of the solitary wave, and its associated flow field, can be

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defined; we consider a wave of permanent form, propagating at a speed c, and so introduce x ctξ = − , with ( , )zφ φ ξ= and ( )η η ξ= , then

2 0zz ξξφ δ φ+ = (2.10)

with 2( )z cξ ξφ δ φ η= − & ( )2 2 210

2 zc ξ ξφ η φ δ φ−− + + + = on 1z η= + (2.11)

and 0zφ = on 0z = . (2.12)

Then we define:

total mass dM η ξ∞

−∞

= ∫ ; total momentum (or impulse) 1

0

d dI zη

ξφ ξ+∞

−∞

= ∫ ∫ ;

total kinetic energy ( )1

2 2 2

0

1d d

2 zT zη

ξφ δ φ ξ+∞

−

−∞

= +∫ ∫ ;

potential energy of the wave 21d

2V η ξ

∞

−∞

= ∫ ; circulation [ ]dC φ∞

∞−∞

−∞

= ⋅ =∫ u s ,

where this last integral is taken along any streamline. As an example of the identities that exist between these various properties of the solitary wave, it is convenient to start from the equation of mass conservation (see (1.10)), written in the moving frame:

( ) 0zu c wξ− + = .

The integration in z, with the relevant boundary conditions imposed, gives

1

0

d( )d 0

du c z

η

ξ

+ − = ∫

and so 1 1

0 0

( )d constant ( )du c z c z cη+

− = = − = −∫ ∫ ,

since both u ξφ= and η tend to zero as ξ → ∞ . Thus

1 1

0 0

d du z z cη η

ξφ η+ +

= =

∫ ∫

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and then 1

0

d d dz cη

ξφ ξ η ξ+∞ ∞

−∞ −∞=∫ ∫ ∫ i.e. I cM= .

This identity was first obtained by Starr (1947); another identity is

2 ( )T c I C= − ,

which was first derived by McCowan (1891). In 1974, Longuet-Higgins found

23 ( 1) .V c M= −

Other results that apply to the solitary wave, up to the largest wave, together with applications to numerical and other approximations, can be found in Longuet-Higgins (1974, 1975), Longuet-Higgins & Fenton (1974) and Longuet-Higgins & Cokelet (1976). [Exercise 4.]

Lecture 2

3 Asymptotic expansions (with a parameter) Our primary interest here is in the representation, as a suitable expansion, of functions that involve a parameter (ε), as that parameter approaches zero. The function will otherwise depend on a variable (which can be a vector); in this introduction to the ideas, we consider the real, scalar function ( ; )f x ε , where x is also a real scalar. The parameter here is ε (real) and we shall always define it so that 0ε > : the limit process is always therefore 0ε +→ i.e. 0ε ↓ (even if we often write simply 0ε → ). The function is defined on some domain, x D∈ , and the critical question is this: is an expansion of ( ; )f x ε , as 0ε → at fixed x, appropriate (valid) for x D∀ ∈ ? First, the expansion of a function, in the asymptotic sense, is constructed with respect to an asymptotic sequence, usually written { ( )}nφ ε , 0,1,2,...n = . The elements

of this sequence are defined so that

[ ]10

lim ( ) ( ) 0n nεφ ε φ ε+→

= for every 0,1,2,...n = ,

usually expressed as

[ ]1( ) o ( )n nφ ε φ ε+ = as 0ε → ,

where we have used the Landau symbol. Then, for a sequence appropriate to a given function, we write

[ ]10

( ; ) ( ) ( ) O ( )N

n n Nn

f x a xε φ ε φ ε+=

= +∑

as 0ε → , at fixed x, for suitable coefficients ( )na x . (We have used the other Landau

symbol here.) This statement is often expressed as

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0

( ; ) ~ ( ) ( )N

n nn

f x a xε φ ε=∑ as 0ε → ,

and read as ‘asymptotically equal to’ or ‘varies as’; we must allow, at least in principle, that N → ∞ (although we are rarely able to construct all the terms explicitly). As implied above, { ( )}nφ ε cannot be arbitrarily assigned; for example, the sequence

{ }nε is appropriate for the function 1 x ε+ + but not for exp( )x ε− . On the other

hand, { ( )}nφ ε is not unique to a given function: for 1 x ε+ + we could also use

{sin }n ε or {tan }n ε or even {ln(1 )}nε+ , but we would normally choose the simplest of various (essentially equivalent) sequences. To proceed, we suppose that ( ; )f x ε is defined for x D∈ ; we construct the

asymptotic expansion of f with respect to the asymptotic sequence { ( )}nφ ε at fixed x,

so that, for every 0N ≥ ,

0

( ; ) ~ ( ) ( )N

n nn

f x a xε φ ε=∑ .

Question: is this asymptotic representation, for N∀ , also valid for x D∀ ∈ ? First, we address the concept of validity in this context. The asymptotic expansion is said to be valid – we normally refer to this as ‘uniformly’ valid – if

[ ]1 1( ) ( ) o ( ) ( )n n n na x a xφ ε φ ε+ + =

for every 0,1,2,...n = and for x D∀ ∈ , as 0ε → . Thus the function

( ; ) 1f x xε ε= + + , 0x ≥ ,

has an asymptotic expansion which starts ( ; ) ~ 12 1

f x xx

εε + ++

;

it is easily checked – here we can construct the terms to all orders – that this asymptotic expansion is uniformly valid on the given domain. Note, however, that this would not be the case if the domain were given as 1x ≥ − ; indeed, the example

2( ; )f x x xε ε= + + , 0x ≥ , has the asymptotic expansion which begins

2

2( ; ) ~

2f x x x

x x

εε + ++

, (2.13)

which is not uniformly valid as 0x +→ (which is in the given domain).

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An asymptotic expansion which is not uniformly valid is said to ‘break down’, this occurring for a particular size (or, possibly, sizes) of x (in the domain); this breakdown is determined by any xs that satisfy

[ ]1 1( ) ( ) O ( ) ( )n n n na x a xφ ε φ ε+ + =

for any n, as 0ε → . In the preceding example, (2.13), these first two terms in the expansion show a breakdown where O( )x ε= (and it is a straightforward exercise to confirm that this is the only breakdown of the complete asymptotic expansion on the given domain). The procedure is to define a new x, of the given size, and repeat the construction of an asymptotic expansion for the new x held fixed as 0ε → . So here, let us set x Xε= (a ‘scaled’ x), to give

2 2( ; ) ( ; ) ( ; ) (1 )f x f X F X X Xε ε ε ε ε ε= ≡ = + + ,

and then we have the corresponding asymptotic expansion which starts

2

( ; ) ~ 12 1

XF X X

X

εε ε + +

+ (2.14)

as 0ε → for X fixed. (You may wish to check that this new asymptotic expansion is valid as 0X +→ , but not as X → ∞ – and then this breakdown simply takes us back to x.) One final comment: the two expansions that we have generated, of the same function, namely (2.13) and (2.14), are said to ‘match’. That is, an appropriate further expansion of (2.13), and correspondingly of (2.14), produce two expressions that are identically the same. We will not spell out the rules here – further details can be found in the references mentioned below – but we will present the relevant calculation for this example. We have

2 2 2

2 2 22 2x x X X

x x X X

ε εε εε ε

+ + = + ++ +

3 21 1 1~

2 42X X X

Xε ε + + −

(2.15)

retaining terms O( )ε and O( )ε ε , and

2 21

1 122 1 1

X xX x

X x

ε εε ε εε

+ + = + + + +

3 21 1 1~ 1 1

2 2 2x x

x x

ε ε + + −

, (2.16)

retaining terms O(1) and O( )ε . The final expressions in (2.15) and (2.16) are identical, when written in the same variable; this is called matching (and this property,

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which also involves the careful designation of the terms to retain, is called the matching principle). A problem that requires more than one (matched) asymptotic expansion, to cover the given domain, is called a singular perturbation problem. [Exercises 5 and 6.] Before we proceed to use these ideas in our studies of the water-wave problem, we make an important observation (which is at the heart of modern singular perturbation theory): we normally do not have a function to expand! Rather, we aim to seek a function which is the solution of a partial (or ordinary) differential equation, or a set of such equations. Thus we need to adjust the approach slightly: we first assume a suitable asymptotic sequence and associated expansion, consistent with the form of the governing equation(s) and boundary conditions. This generates a sequence of problems, obtained at each order, which are then solved sequentially. Once we have an asymptotic solution constructed in this way, valid in some domain, we may examine its validity: is it valid throughout the domain of interest? Is there a breakdown requiring a rescaling of the variable(s)? (The interested reader may wish to explore these ideas further; a selection of relevant texts is provided by van Dyke (1964), Copson (1967), Smith (1985), Holmes (1995), Kevorkian & Cole (1996) and Johnson (2004).) Of course, this approach is based on a ‘formal’ procedure: it simply follows a well-defined path, guided by clear principles. It does not incorporate any element of rigour, in the sense of proving uniqueness – typically, rather unimportant in this context because solutions are usually known to be unique – and existence. This latter point can be viewed on two levels. On the one hand, we need to be satisfied that a solution exists within a certain class of functions. This is not quite so critical here, because the asymptotic development almost always leads to the specification of the functions that are allowed, and readily identifies failures (which may then, if appropriate, be examined via scaling and matching). On the other hand, there is a far more significant issue relating to the nature of the asymptotic expansion itself: does it exist in the sense of constituting a convergent series with a non-zero radius of convergence? Sadly, the answer is very often ‘no’ in the context of classical singular perturbation theory. Asymptotic expansions, as described above, are, in general, strictly only asymptotic: they may well diverge when viewed conventionally. One of the aims of a rigorous approach is to prove that, for some 0 0ε > , the expansion converges for 00 ε ε≤ < (at

least for a certain class of functions). This can sometimes be accomplished, but typically with other constraints e.g. in wave-propagation problems – particularly relevant here – convergence often requires 0τ τ< for some 0τ fixed as 0ε → , where τ

is a scaled time ( tτ ε= ), and for suitably smooth initial data. (That such a solution is not valid for τ → ∞ is not normally a significant difficulty. It is understood that such a problem, in all likelihood, has been generated from physical considerations, and has already ignored some physical properties, e.g. viscosity, which become important,

typically on timescales longer than 1O( )ε − : we do not expect our asymptotic solution to be appropriate for τ → ∞ .) Without the bonus of convergence, our asymptotic expansions are, therefore, no more than formal representations of the solution, expressed as divergent series – but this is not quite as worrying as it might appear. The fundamental difference between divergent and convergent series is simply stated: the error in using a convergent series can be made vanishingly small by

14

increasing the number of terms indefinitely. In the case of a divergent series, for a given x and 0ε > (where we use our notation introduced above), there is a choice of N (the number of terms) which minimises the error; this is the best that we can accomplish. Nevertheless, even if only numerical estimates are required, this still gives a very useful (and mathematically robust) result; the real strength is that the detailed mathematical (i.e. algebraic/functional) structure of the solution is obtained by these techniques. (For those who wish to explore these ideas further, a very illuminating discussion of the relevance and use of divergent series can be found in Hardy (1949); see also Ford (1960).) We submit, therefore, that the asymptotic approach can certainly contribute to a clearer understanding of complicated phenomena – but, of course, it can never compete with rigorous mathematics. It may be the spring-board for a rigorous development, or it may attempt to fill-in some of the detail not accessible from the rigorous approach. With these short comings in mind, we now use these ideas to investigate a number of important and illuminating problems that are generated by the classical water-wave equations.

4 Weakly nonlinear, weakly dispersive waves As an introduction to our discussion, and to lay the foundations for much that we shall present here, we will begin by deriving the Korteweg-de Vries (KdV) equation as it appears in water waves. This will provide the background to the derivation of related equations, the majority of which turn out to be integrable equations (in the soliton or inverse scattering sense; see Chapter 5). These equations represent asymptotic approximations to the water-wave problem, exhibiting some appropriate balance between nonlinearity and dispersion. Although many of these equations are useful in the description of important properties of water waves, the emphasis here will be on the method of derivation (and we can also note the existence of a surprising number of interesting equations that are generated by this one problem). 4.1 The Korteweg-de Vries (KdV) equation This is arguably the most important equation of this type in this context. It was the first to be obtained (originally by Korteweg and de Vries, 1895) – although we will give a more modern derivation of it – and it can be regarded as the archetype for the study of soliton problems. We choose to start with equations (1.10)-(1.12), with 0b ≡ and (1.16) incorporated, and we consider only plane waves propagating in the X-direction, where ( , )X Y⊥ =X

and ( ,0)u⊥ =u ; there will be no dependence on Y. (We use the Euler equation even

though we will assume irrotationality here, because we will then be able to compare this with the calculations with vorticity (examined later).) So we introduce X Tξ = − , Tτ ε= , (4.1) which describes a far field for right-running waves. [The corresponding near field

arises with the choice X Tξ = − and T (i.e. O(1)T = , not 1O( )T ε −= as used in (4.1)); this generates a perturbed linear problem, which produces an asymptotic

15

solution not uniformly valid for 1O( )T ε −= ; we have therefore gone directly to the far field. Typically, these problems produce a solution in the far field which is uniformly valid as 0τ → .] [Exercise 7.] With this choice of variables, the governing equations become

( )zu u uu Wu pξ τ ξ ξε− + + + = − ; { }( )z zW W uW WW pξ τ ξε ε− + + + = − ; 0zu Wξ + = ,

(4.2) with p η= & ( )W uξ τ ξη ε η η= − + + on 1z εη= + (4.3)

and 0W = on 0z = . (4.4) In these equations, we have again introduced subscripts to denote partial derivatives. We seek a solution of this set for [0,1 ]z εη∈ + , ξ−∞ < < ∞ and 0τ ≥ , and for suitable initial data. The procedure now is to assume a formal asymptotic solution of the system (4.2)-(4.4), expressed as

0

( , ; ) ~ ( , )nn

n

η ξ τ ε ε η ξ τ∞

=∑ and

0

( , , ; ) ~ ( , , )nn

n

q z q zξ τ ε ε ξ τ∞

=∑ , (4.5)

where q (and correspondingly nq ) represents each of u, W and p. These asymptotic

expansions are, typically, uniformly valid as 0ε → , for 00 τ τ≤ < , for some fixed 0τ ,

provided that the initial data decays sufficiently rapidly as ξ → ∞ . Indeed, matching

to the near field (where O(1)T = , and containing 0T = ) shows that the problem in the far field must satisfy the initial data prescribed at 0T = ; this will take the form of some given ( ,0)Xη , but this function will have to decay sufficiently rapidly at infinity in order to ensure the existence of solutions of the KdV equation. Thus this far-field solution will be uniformly valid back into the near field. (Periodic solutions require a more careful treatment.) The procedure is altogether routine; at leading order we obtain

0 0p η= , 0 0u η= , 0 0W zη= − (all for 0 1z≤ ≤ ),

where we have imposed the condition that the perturbation in u is caused only by the passage of the wave i.e. 0 0u = whenever 0 0η = . Because the boundary conditions are

automatically satisfied, the leading-order contribution to the surface wave (0η ) is

arbitrary – we must go to the next order to determine it. Evaluation on the free surface is, at this order, simply 1z = ; to proceed, we define all the functions on [0,1]z∈ and so we invoke Taylor expansions (of the surface boundary conditions) about 1z = . This is a valid manoeuvre here because the solution, at all orders, turns out to be polynomial in z, which ensures that the asymptotic solution is certainly uniformly valid in z. (Indeed, the construction of a power-series solution in z was the basic structure invoked by Korteweg and de Vries in 1895.) Thus the two surface boundary conditions, (4.3), are written as

16

( )

20 0 0 1 0 1

20 0 0 1 0 1 0 0 0

O( )

O( )

z

z

p p p

W W W uξ ξ τ ξ

εη ε η εη ε

εη ε η εη ε η η ε

+ + = + +

+ + = − − + + +

on 1z =

together with, at O(ε),

1 0 0 0 0 0 1zu u u u W u pξ τ ξ ξ− + + + = − ; 1 0zp W ξ= ; 1 1 0zu Wξ + = .

These equations and boundary conditions then give, for example,

{ }210 1 02

~ (1 )p z ξξη ε η η+ + −

and

( ){ }31 10 1 0 0 0 0 02 6

~W z z zξ ξ τ ξ ξξξ ξξξη ε η η η η η η− + − + + + + ,

both defined for 0 1z≤ ≤ . In order to satisfy the surface kinematic boundary condition, the function 0( , )η ξ τ satisfies the Korteweg-de Vries (KdV) equation

10 0 0 03

2 3 0τ ξ ξξξη η η η+ + = , (4.6)

(see Exercise 8) leaving the next approximation to the free surface, 1η , undetermined

at this order; the equation for this function is found by going to the next order (see Exercise 9). The KdV equation provides the basis for a discussion of a class of water waves, which includes the solitary wave and soliton interactions; the evidence (see, for example, Constantin & Johnson (2008a)) is that, on suitable distance and time scales, this is a good model for nonlinear, dispersive water waves. Indeed, the scales on which the KdV equation might be relevant is of some interest (if only because there has been some debate over the relevance of the KdV, and its soliton solutions, to the formation and development of tsunami waves). The distance scale (and corresponding time scale) which dictates where the KdV equation is

appropriate is given by 1O( )X ε −= i.e. 3 2O( )x δ ε= (which is obtained from (1.16) and (4.1)); this produces a length scale measured by

3 2

03 20

ah

h

δλε

−

=

( 0a h< ),

which depends only on the (average) amplitude and (average) depth. This can be used to estimate the distance at which the KdV balance occurs, for various depths and amplitudes:

17

amplitude: 0 1⋅ m 0 5⋅ m 1m 5m 10m depth 5 m 1 77⋅ km 0 16⋅ km 56 m N/A N/A depth 10 m 10 km 0 89⋅ km 0 32⋅ km 28 m N/A depth 100 m 3162 km 283 km 100 km 8 9⋅ km 3 2⋅ km depth 1 km 610 km 48 9 10⋅ × km 43 2 10⋅ × km 2828 km 1000 km

depth 4 km 73 2 10⋅ × km 72 8 10⋅ × km 610 km 49 10× km 43 2 10⋅ × km

Table showing the distances for the KdV balance, for various depths and amplitudes

We observe that, for moderate amplitudes in a depth typical of rivers, the balance occurs over a few kilometres which makes it possible (as J. Scott Russell did in 1834) to follow a solitary wave on horseback; see Russell (1844). On the other hand, the 2004 Boxing Day tsunami (for which the initial a was barely 1m and the local ocean depth 4km) requires distances of about one million kilometres! This tsunami was not of soliton/KdV origin; see Constantin & Johnson (2008b).

Lecture 3 4.2 The two-dimensional Korteweg-de Vries (2D KdV) equation The KdV equation obtained in the previous section describes nonlinear plane waves propagating, in the X-direction, on stationary water. We now pose the question: how might this equation be modified if the propagation is on the two-dimensional surface? A simple example would be the oblique interaction of two waves that, at infinity, were plane but not parallel. However, we aim to derive an equation that is essentially a KdV-type equation and, hopefully, also completely integrable; this requires the additional dependence (on Y) to be special i.e. appropriately ‘weak’. One way to see what is involved is to formulate, initially, the near-field, linear, long-wave equation for this problem: ( ) 0TT XX YYη η η− + = , (4.7)

to leading order as 0ε → ; this is the classical, two-dimensional wave equation. A solution of this equation is

i( )e kX lY Tωη + −= where 2 2 2k lω = + ,

and we consider l small: 2

2

1~ 1

2

lk

kω

+

as 0l → ; this then gives the dispersion

relation for waves that are propagating predominantly in the X-direction. In order that the contribution from the behaviour in the Y-direction is the same size as the nonlinearity and dispersion already present in the KdV equation, we must choose

O( )l ε= . However, this is more conveniently accommodated by rescaling according

to Yεϒ = and then, for consistency with the equation of mass conservation (i.e. for

18

the existence of a suitable stream function) we also require v Vε= (where ( , )u v⊥ =u ).

The governing equations, again written in far-field (x-)variables ξ and τ, then become

( )zu u uu Vu Wu pξ τ ξ ξε ε ϒ− + + + + = − ; ( )zV V uV VV WV pξ τ ξε ε ϒ ϒ− + + + + = − ;

{ }( )z zW W uW VW WW pξ τ ξε ε ε ϒ− + + + + = − ; 0zu V Wξ ε ϒ+ + = ,

with p η= & ( )W u Vξ τ ξη ε η η ε ηϒ= − + + + on 1z εη= +

and 0W = on 0z = .

The asymptotic expansion, and general procedure, follow exactly as for the KdV derivation, producing now, for example,

( ){ }31 10 1 0 0 0 0 0 02 6

~W z V z zξ ξ τ ξ ξξξ ξξξη ε η η η η η ηϒ− + − + + + + + ,

and then finally the equation for the leading-order term representing the wave is given as:

( )10 0 0 0 03

2 3 0τ ξ ξξξ ξη η η η η ϒϒ+ + + = . (4.8)

This equation, (4.8), is the two-dimensional KdV equation (often referred to as the Kadomtsev-Petviashvili (KP) equation; see Kadomtsev & Petviashvili (1970)), and it is another completely integrable equation (see Chapter 5). It turns out that there are two variants of this equation, both of some interest, defined by 0η ϒϒ± ; the equation

with our sign – the case applicable to water waves – is usually designated KPII. It is instructive to give an interpretation of the scalings that we have used in the derivation of the 2D KdV equation, (4.8). This equation is an appropriate leading-order approximation, valid at times O(1)τ = (so the original non-dimensional, scaled time,

T, is large: 1O( )ε − ) and where O(1)ξ = (which is a measure of the width of the

wave). However, the solution is also expressed in terms of Yεϒ = , which can be regarded as a ‘weak’ dependence on Y. Thus, along any wave front, we have

d d O( )Y X ε= : in the original, non-dimensional variables, ( , )X Y for some

O(1)τ = , the wave front deviates only a little ( )O( )ε away from a plane wave

(defined by lines constantξ = ). So, for example, in the case of two obliquely crossing waves – an exact solution of the 2D KdV equation (see Chapter 5) – the angle between the waves in ( , )X Y or ( , )Yξ coordinates is small: the two waves are nearly parallel in the original physical frame (and they propagate in essentially the same direction: head-on collisions are not allowed; see §4.5).

19

4.3 The concentric (or cylindrical) Korteweg-de Vries (cKdV) equation We have derived the versions of the Korteweg-de Vries, in both one and two dimensions, as described in Cartesian geometry; we now demonstrate that a corresponding equation exists in cylindrical coordinates. The same general approach is possible, but some elements of the formulation are different in important details. To see one of the main differences, it is useful first to consider the underlying linear problem for long waves in this geometry:

1

0tt rr rrη η η − + =

; (4.9)

cf. equation (4.7). Here, r is the radial coordinate, non-dimensionalised with respect to the wave length (λ) and, at this stage, without the additional scaling described in (1.16). The solution of (4.9), for outward propagation and at large radius, is

1~ ( )f r t

rη − (as r → ∞ , for O(1)r t− = );

this shows the expected geometrical decay of the amplitude of the wave, as the radius increases, and this will need to be incorporated in any scaling that we adopt. The relevant governing equations, written in cylindrical coordinates, but with circular symmetry, are

( )t r z ru uu wu pε+ + = − ; ( ){ }2t r z zw uw ww pδ ε+ + = − ;

10r zu u w

r+ + = ,

with p η= & t rw uη ε η= + on 1z εη= +

and 0w = on 0z = ; cf. (1.10)-(1.12). There is a scaling, equivalent to (1.16), that replaces the two parameters, ε and δ, by a single parameter; we introduce

2

2( )r t

εξδ

= − and 6

4R r

εδ

= ,

and define 3

2H

εηδ

= , 3

2p P

εδ

= , 3

2u U

εδ

= , 5

4w W

εδ

= ,

which describes outward propagation at a suitable radius. The governing equations then become

( ) ( )z R RU UU WU UU P Pξ ξ ξ− + ∆ + + ∆ = − + ∆ ;

20

( ){ }z R zW UW WW UW Pξ ξ∆ − + ∆ + + ∆ = − ; 1

0z RU W U URξ

+ + ∆ + =

,

with P H= & ( )RW H UH UHξ ξ= − + ∆ + ∆ on 1z H= + ∆

and 0W = on 0z = ,

where 4 2ε δ∆ = is the single parameter that appears in this system. Thus the problem associated with 0∆ → can be interpreted as, for example, 0ε → at fixed δ (and the amplitude parameter is now defined as that appropriate to the region where O(1)r = and O(1)t = ); the amplitude in the far field is ∆. Indeed, with O(1)ξ = (the neighbourhood of the wave front) and O(1)R = , where

( )( ) ( )4 2 2 2 2 2R r rε δ ε δ ε δ= = ∆ ,

then 0∆ → implies 2 2rε δ → ∞ : this scaling corresponds to large radius. The procedure is exactly as in the two previous derivations; an asymptotic solution is sort, as presented in (4.5), but now the expansion parameter is ∆. At leading order, we obtain

0 0P H= , 0 0U H= , 0 0W zH ξ= − (0 1z≤ ≤ ),

for arbitrary 0( , )H Rξ ; at the next order, we find, for example,

1 1 0 01

z RW U U URξ

= − − +

(4.10)

and then we get the equation for 0( , )H Rξ :

0 0 0 0 01 1

2 3 03RH H H H H

R ξ ξξξ+ + + = , (4.11)

which should be compared with the classical KdV equation, (4.6). This is the concentric KdV equation which is also a completely integrable equation (even if the variable coefficient might suggest otherwise!). 4.4 The nearly-concentric Korteweg-de Vries (ncKdV) equation We have seen how the classical one-dimensional KdV equation (§4.1) has a two-dimensional counterpart (with weak y-dependence): the 2D KdV equation (§4.2). The same general property arises with the cKdV equation; a large radius wave – the far-field – can have, in addition, a weak dependence in the θ direction (when written in cylindrical coordinates). The wave equation for linear, long waves, in cylindrical geometry, is

2

1 10tt rr rr r

θθη η η η − + + =

;

21

the far field, with a suitable weak θ dependence, requires the introduction of the

additional scaled variable defined by ( )2θ ε δ= Θ , together with ( )5 3v Vε δ=

(where ( , )u v⊥ =u in cylindrical polar coordinates). The procedure is exactly as

described for the cKdV equation (and the leading order is identical) but now we have, for example,

( )1 1 0 0 01

z RW U U U VRξ Θ= − − − + ;

cf. equation (4.10). The ncKdV equation appears at this order:

0 0 0 0 0 021 1 1

2 3 03RH H H H H H

R Rξ ξξξ

ξΘΘ

+ + + + =

, (4.12)

but this is not a completely integrable equation for general initial data; see §4.8. 4.5 The Boussinesq equation The examples discussed so far describe propagation in one direction only – either to the right (or left) or outwards. We now consider the possibility of a model that allows waves to propagate to the left and to the right, both being solutions of the same equation (which, of course, is the situation pertaining to the original, governing equations). This will allow waves to collide head-on. We shall find, however, that allowing both together – so we cannot follow either one or the other (cf. §4.1 and Exercise 8) – implies that the equation we require must allow propagation predominantly in either direction, with weak nonlinearity and weak dispersion incorporated as small corrections. We start with our standard equations for one-dimensional propagation, (1.10)-(1.12) with (1.16): ( )T X z Xu uu Wu pε+ + = − ; { }( )T X z zW uW WW pε ε+ + = − ; 0X zu W+ = ,

with p η= & T XW uη ε η= + on 1z εη= +

and 0W = on 0z = . We seek a solution in the standard form, (4.5), and then, at leading order, we find that

0 0p η= ; 0 0T Xu η= − ; 0 0XW zu= − ; 0 0X Tu η= − (all for 0 1z≤ ≤ ),

and so 0 0 0TT XXη η− = . (4.13)

As before, we expand the surface boundary conditions about z = 1; at the next order we then obtain

211 0 12

(1 ) XTp z u η= − − + ; { } 31 11 0 0 1 0 02 6

( )T X X XX XXXT XXXTW u u u z z uη= + − + ,

22

and eventually

10 0 1 0 0 0 1 0 03

( ) ( ) ( )X X XX XXXT X T TT X Tu u u u uη η η η+ − − = + . (4.14)

Finally, we invoke the leading order (in the form 0 0 dTX

u Xη∞

= ∫ , on the assumption

that decay conditions exist ahead of the wave i.e. the wave is entering undisturbed

conditions) and construct the equation for 20 1 O( )η η εη ε= + + :

2

2 21 1d O( )

2 3TT XX T XXXXXXX

Xη η ε η η εη ε∞ − − + − =

∫ . (4.15)

This is the Boussinesq equation (Boussinesq, 1871) and, in this context, it is valid for X = O(1) and T = O(1), combining the O(1) and O(ε) (and ignoring the error term, of course). It admits solutions that describe waves that propagate to the left and to the right, together, but with a weak nonlinear interaction between them and for weak dispersion. However, in suitable far fields, equation (4.15) recovers the KdV equations for, separately, right- or left-propagating waves; see Exercise 10. We comment that this type of equation is the least satisfactory outcome when we use this asymptotic structure: there are terms of different asymptotic order (here O(1) and O(ε)) retained together in the same equation; cf. the KdV equation, (4.6). This equation, (4.15), is another of our completely integrable equations, although the form in which it is written does not allow immediate solution by this method. To

do this, we must transform the equation (and take advantage of the error: 2O( )ε here). We introduce

2H η εη= − and ( , ; )dX

X X T Xχ η ε∞

′ ′= − ∫ , (4.16)

which is equivalent to writing the equation in a Lagrangian rather than an Eulerian frame; this gives

2 23 12 3

( ) O( )TTH H H Hχχ χχ χχχχε ε ε− − − = ; (4.17)

this is the Boussinesq equation written in completely integrable form. Complete integrability holds for any ε > 0 (in the absence of the error term); solutions generated, however, are relevant only in the region defined by T = O(1) and χ = O(1), with H = O(1). With this understanding, we may scale

2, ( , ) ( , )

3H H T T

εχ χε

→ − →

to recover the Boussinesq equation written in the standard form:

23

23( ) 0TTH H H Hχχ χχ χχχχ− + − = . (4.18)

This equation possesses solutions, and most particularly soliton solutions, that may propagate either all in the same direction (much as for the KdV equation) or in opposite directions, producing head-on collisions between the waves; see §5.4. 4.6 The two-dimensional Boussinesq equation The KdV equation, as we have seen, can be generalised to accommodate a weak dependence in the y-direction, and so admit solutions representing obliquely interacting waves; see §4.2. The same approach can be adopted for the Boussinesq equation, by allowing a weak dependence in y and so this will model the oblique, head-on collision of waves. The procedure follows the method of derivation used for the 2D KdV equation, by extending the development that leads to the Boussinesq equation (§4.5) with the inclusion of a suitable y-dependence.

So we define a scaled y, by writing Yεϒ = and then, for consistency with the

equation of mass conservation, introducing v Vε= (where ( , )u v⊥ =u ); see §4.2.

We obtain – perhaps not surprisingly – the 2D Boussinesq equation

2 23 12 3

( ) O( )TTH H H H Hχχ χχ χχχχε ε ε εϒϒ− − − − = , (4.19)

written in the same Lagrangian frame as for equation (4.17). This, in turn, can be expressed in the standard form for an integrable system, by transforming

2, ( , ) ( , ), 3

3H H T T

εχ χ εε

→ − → ϒ → ϒ ,

to produce

23( ) 0TTH H H H Hχχ χχ χχχχ ϒϒ− + − − = . (4.20)

As we shall mention later, this equation has some interesting properties that relate to integrability, but it falls short of being completely integrable. More details about this equation, its solutions and relevance, can be found in Johnson (1996).

Lecture 4 4.7 The Camassa-Holm (CH) equation This equation is the last that we shall discuss under the heading of weakly nonlinear, long waves i.e. weakly dispersive waves, but it is by the far most involved in terms of its derivation. Furthermore, it has the same asymptotic character as the Boussinesq equation: we retain terms of different orders in the one equation. Indeed, this is further complicated by the need to use both our fundamental small parameters (ε and δ) independently and in the same equation. Relevant background information can be found in Camassa & Holm (1993).

24

We start with equations (1.10)-(1.12), with both ε and δ retained as separate and independent parameters (with the understanding that 0ε → , 0δ → ); we consider one-dimensional wave propagation (in the x-direction). The method that we describe here is explained more fully, particularly in the context of other model equations for water waves, in Johnson (2002). First, we introduce

( ), ,x t t w wζ ε τ ε ε ε= − = = ,

for right-running waves (and the geometry is restricted to two dimensions, (x, z)); we

form the equation for 2 200 10 01 11( , ; , ) ~η ζ τ ε δ η εη δ η εδ η+ + + , which gives

2 2 2 2 431 13 4 12

2 3 (23 10 ) O( , )τ ζ ζζζ ζ ζ ζζ ζζζη ηη δ η εη η εδ η η ηη ε δ+ + − = − + + ,

(4.21) but this is not a CH equation – although it does take the form of higher-order corrections to the KdV equation. The details underpinning this derivation are quite straightforward, following the general approach adopted for the KdV equation, but working to higher order. Here, we can interpret the procedure as: expand first in ε (so each term in this asymptotic expansion also depends on δ), and then expand each of these terms in δ. We elect to retain terms as far as O(ε) in the first expansion, and then each of the O(1) and O(ε) terms are themselves expanded, to retain terms as far as

2O( )δ . (A comment on a rigorous justification of this approach will be made later.) To proceed, we find that, to this same order of approximation, we have

( )2 2 21 1 14 3 2

~u z ζζη εη εδ η− + − (for 0 1z≤ ≤ ), (4.22)

which describes the horizontal velocity component in the flow at various depths. We

now select a specific depth, denoted by 0z z= ( 00 1z≤ ≤ ) and introduce 21 103 2 zσ = − ;

writing 0ˆ ( , , ; , )u u zζ τ ε δ= , we invert (4.22), evaluated at 0z z= , to give

2 214

ˆ ˆ ˆ~ u u uζζη ε εδ σ+ − . (4.23)

This is used in (4.21) to obtain the corresponding equation in u :

{ }2 2 2 429 513 12 6

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2 3 (6 ) O( , )u uu u u u uuτ ζ ζζζ ζ ζζ ζζζδ εδ σ ε δ+ + = − + + + ,

or

{ }2 2 2 3 429 51ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ3 12 6

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2( ) 3 (6 ) O( , )x x xxx x xx xxxtu u uu u u u uuε εδ ε δ σ ε εδ+ + + = − + + + ,

(4.24)

when expressed in (essentially) original variables: ˆˆ ,x x t tε ε= = . Finally, we add

the term 2ˆ ˆˆ ˆ ˆ ˆˆ ˆ( )xxt xxtu uεδ µ − to the left-hand side of (4.24), and use

25

3ˆ ˆ ˆ2

ˆ ˆ ˆ ˆ~ ( )x xtu u uuε− +

in the first of these; finally, we choose 1

2 4µ σ= + with 56µ = (so that 1

12σ = ), to give

{ }2 2 2 2 3 45 51ˆ ˆˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆˆˆ ˆ2 6 12

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2( ) 3 2 O( , )x x xxx x xx xxxt xxtu u uu u u u u uuε εδ εδ ε δ ε εδ+ + − − = + + .

(4.25) This is a CH equation (because a simple frame shift allows the term ˆ ˆ ˆˆxxxu to be

subsumed into ˆˆ ˆˆxxtu ), and then the reversion to the standard form requires no more than

a simple scaling transformation; so 5 33 5

ˆ ˆˆ2 ( )x tζ = − , 5

ˆ ˆ3

u u→ (and t unchanged)

gives

{ }2 2 2ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2 3 2t t

u u uu u u u uuζ ζ ζζ ζ ζζ ζζζκ ε εδ ε δ+ + − = + , (4.26)

at this order, where (2 5) 3 5κ = . We have demonstrated that the Camassa-Holm

equation does indeed describe a class of water waves, but this description applies only

to the horizontal velocity component in the flow, at a specific depth (0 (1 2)z = ), and

then, most importantly, if we retain (when expressed in ( , )ζ τ variables; see (4.21))

only terms O(1), O( )ε , 2O( )δ and 2O( )εδ . The equation for the surface, (4.21), as we have previously observed, is not a CH equation; the behaviour of the surface (at this order) is obtained from (4.23) with u determined by (4.26). It should be noted that any number of different equations, all valid at this same level of approximation, are clearly possible, by using the lower-order terms in various higher-order terms (and incorporating additional terms, suitably approximated, as we did with ˆˆ ˆˆxxtu ). In no

sense is the CH equation unique, but it is certainly one of the possibilities, and to show that this was the case was the main aim of the exercise. Of course, we prefer to generate equations like CH because they are completely integrable and, in this context, they incorporate more effects associated with water-wave propagation than, say, just the KdV equation. This analysis and derivation, and the existence of the CH equation as a model for water waves, has been put on a rigorous basis by Constantin & Lannes (2009). There, it is shown that the CH equation for the horizontal velocity component, at the depth that we have found, is a proper approximation of the water-wave problem provided that Mε δ≤ , for some 0M > (independent of ε and δ) and 0δ δ< for some 0 0δ > ,

with an overall error 4O( )εδ in equation (4.26). This, it is shown, constitutes a well-

posed problem, and that there is some fixed 0T ensuring that the solution exists for

times 0ˆ [0, ]t T ε ε∈ .

We conclude by observing that there is a second equation, of CH type, that is also completely integrable: the Degasperis-Procesi equation (Degasperis & Procesi, 1999; Degasperis, et al., 2002)

26

{ }2 2 2ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2 4 3t t

u u uu u u u uuζ ζ ζζ ζ ζζ ζζζκ ε εδ ε δ+ + − = + ,

which we have written in the same format as our CH equation, (4.26). This arises in the water-wave problem in exactly the same way that CH does, but at a depth

1 110 3 2

z = with 32140 10

2κ = ; see Johnson (2003c). Indeed, it is also possible to

obtain a 2D CH equation (y yε= , v v ε= ); the resulting equation that corresponds to (4.26) is then

( ) ( )2 2 2 2ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2 3 2yyt tu u uu u u u u uuζ ζ ζζ ζ ζζ ζζζς ςκ ε εδ ε ε δ+ + − + = + ;

see Johnson (2002). 4.8 Transformations between these equations Before we move on, we make one further general observation about some of the equations described above. The four equations: KdV (4.6), 2D KdV (4.8), cKdV (4.11) and the ncKdV (4.12), are different representations of the same type of propagation problem – weakly nonlinear, weakly dispersive gravity waves on the surface of water – but written in different coordinate systems. Because we may interpret the two equations in cylindrical geometry as being appropriate for large radius, which presumably corresponds to nearly plane waves, we might expect some transformations to exist between these equations.

Thus we see that the 2D KdV equation (4.8), expressed in the form 0 ˆ( , )Hη ξ τ= ,

with 2ˆ 2ξ ξ τ= + ϒ , gives (after one integration in ξ and invoking decay conditions at infinity)

ˆ ˆ ˆ ˆ1 1

2 3 03

H H HH Hτ ξ ξξξτ+ + + = .

This is precisely the cKdV equation, (4.11), expressed here in terms of τ rather than R.

Similarly, in the case of the ncKdV equation, (4.12), we write 0ˆ ˆ( , , )H Rη ζ= ϒ , where

212

ˆ Rζ ξ= − Θ and ˆ Rϒ = Θ , to give

( )1ˆ ˆ ˆ ˆ ˆ ˆ3 ˆ

2 3 0R ζ ζζζ ζη ηη η ηϒϒ+ + + = ,

which is the 2D KdV equation, (4.8), expressed in terms of R rather than τ. Thus, although the ncKdV equation is not completely integrable (for arbitrary initial data), this equation can be solved by soliton methods for initial data consistent with the above transformation (because the 2D KdV equation is completely integrable).

27

5 The inverse scattering transform (‘soliton’ theory) The development of what we now call the inverse scattering transform method came out of some observations of the numerical solution of a related problem (Fermi, et al., 1955) and then of the KdV equation (Zabusky & Kruskal, 1965). The results were a considerable surprise: waves that interacted nonlinearly but retained their identities. This led a group at Princeton to investigate the properties of the equation, which produced (Gardner, et al., 1967) a method of solution that treats the unknown function (a solution of KdV) as the time-dependent potential in a one-dimensional linear scattering problem. This linear problem, with the associated inverse scattering problem, coupled with the time evolution of the potential (consistent with the KdV equation), produce a solution-method that maps the nonlinear PDE into a linear integral equation. This constitutes the inverse scattering transform method (and is somewhat analogous to the classical transform methods for solving linear PDEs). From this relatively small – and very specialised – beginning has sprung a whole range of methods which are applicable to an extremely large family of important equations. In particular, we now have a good understanding of the underlying properties that lead to complete integrability, and many different techniques for constructing solutions. In these lectures, we will do no more than give the simplest statement of the formulation of the solution of a few equations via an integral-equation problem. (This will not include any development of the method itself, and certainly not the description in terms of a quantum scattering problem.) The topic has expanded to include Hamiltonian structure, conservation laws, Bäcklund transforms, Hirota’s bilinear form, prolongation structures, Painlevé equations and much more; the interested reader is directed to e.g. Ablowitz & Segur (1981), Ablowitz & Clarkson (1991), Drazin & Johnson (1992). 5.1 The Korteweg-de Vries equation The solution, ( , )u x t , of the KdV equation

6 0t x xxxu uu u− + = ,

which has been written in the standard form (by simply applying a suitable scaling transformation to any other equivalent KdV equation), is related to the function

( , ; )K x z t by d

( , ) 2 ( , ; )d

u x t K x x tx

= − .

(The wave of elevation then corresponds to u− .) Here, K satisfies the integral equation

( , ; ) ( , , ) ( , ; ) ( , , ) d 0x

K x z t F x z t K x y t F y z t y∞

+ + =∫ ,

which is usually called the Marchenko equation (or, sometimes, the Gel’fand-Levitan equation). In this equation, ( , , )F x z t satisfies the pair of linear equations

28

0xx zzF F− = , 4( ) 0t xxx zzzF F F+ + = ;

however, because we require evaluation on z x= (in order to recover u), the relevant solution for F depends on (x + z) and so this pair reduces to the single equation

8 0tF Fξξξ+ = where x zξ = + .

With this choice, the integral equation becomes

( , ; ) ( , ) ( , ; ) ( , ) d 0x

K x z t F x z t K x y t F y z t y∞

+ + + + =∫ .

This formulation is appropriate for solving the initial-value (Cauchy) problem for the KdV equation, provided that certain existence conditions are satisfied:

( , ) du x t x∞

−∞< ∞∫ and (1 ) ( , ) dx u x t x

∞

−∞+ < ∞∫ for t∀ .

These state that the solution, for all time, must satisfy absolute integrability and, indeed, it must decay sufficiently rapidly at infinity (and exponential decay is what we usually encounter). The simplest solution is obtained by choosing the solution for F as

( )e k x z tF ω α− + + += ,

where k (> 0) is a parameter and α an arbitrary constant (equivalent to writing ( )e k x z tF A ω− + += ) and ( )kω – the dispersion relation – is to be determined. We find

that 38kω = , and then the solution for K follows directly:

( ) 38

e( , ; )

e 2 e

kz

kx kx k tK x z t

k α

−

− − −=

+;

thus we obtain a solution of the KdV equation as

2 2 30( , ) 2 sech ( ) 4u x t k k x x k t = − − −

( 022 e e kxk α −− = ).

This is the solitary-wave solution of the KdV equation, and corresponds to the small-amplitude solitary wave discussed in §2.2. The two-soliton solution is generated by the choice

1 2e eF θ θ= + where 3( ) 8i i i ik x z k tθ α= − + + + , 1 2k k≠ ,

29

which eventually leads to the solution

( )

2 2 2 21 1 2 2 1 2 1 2 2 1 1 2 1 2

21 2 1 2

2( ) ( )( , ) 8

1

k E k E k k E E A k E k E E Eu x t

E E AE E

+ + − + += −+ + +

,

where

3exp 2 ( ) 8i i oi iE k x x k t = − −

(i = 1, 2) and 2 21 2 1 2( ) ( )A k k k k= − + ;

here, 0ix are two arbitrary phase shifts. An example of a two-soliton solution is shown

in Figure 2. [Exercises 11, 12, 13.]

Fig. 2 Perspective view of a 2-soliton solution of the Korteweg-de Vries equation.

Lecture 5

5.2 The two-dimensional Korteweg-de Vries equation The solution of the 2D KdV equation, written in the form

( )6 3 0t x xxx yyxu uu u u− + + = ,

can be expressed as

d( , , ) 2 ( , ; , )

du x t y K x x t y

x= − ,

where

( , ; , ) ( , , , ) ( , ; , ) ( , , , ) d 0x

K x z t y F x z t y K x y t y F y z t y y∞ ′ ′ ′+ + =∫ .

The function F satisfies the pair of equations

30

0xx zz yF F F− − = , 4( ) 0t xxx zzzF F F+ + = ,

and then the solitary-wave solution is obtained with the choice

2 2 3 3exp ( ) ( ) 4( )F kx lz k l y k l t α = − + + − + + +

;

cf. the solution of KdV, §5.1. The solution based on the sum of two exponentials for F produces the two-soliton solution (i.e. an oblique collision), an example of which is shown in Figure 3. [Exercise 14.]

Fig. 3 A 2-soliton solution of the 2D Korteweg-de Vries equation.

5.3 The concentric Korteweg-de Vries equation We write this equation in the standard form

16 0

2t x xxxu u uu ut

+ − + = ,

and the Marchenko equation is unchanged:

( , ; ) ( , , ) ( , ; ) ( , , ) d 0x

K x z t F x z t K x y t F y z t y∞

+ + =∫ ,

but where F now satisfies the pair of equations

( )xx zzF F x z F− = − and 3 t xxx zzz ztF F F F xF zF− + + = + .

In this case, the transformation that recovers the solution is

31

( ) 2 3 d( , ) 2 12

d

Ku x t t

η−= − where ( , ; )K K tη η= with 1 3(12 )x tη = ,

and it is this appearance of a similarity variable which complicates the solution somewhat! [Exercise 15.] 5.4 The Boussinesq equation The soliton solutions of the Boussinesq equation, written as

23( ) 0tt xx xx xxxxu u u u− + − = ,

can be recovered from 2

22 lnu f

x

∂= −∂

, for suitable ( , )f x t . The solitary-wave

solution is then given by the choice ( )1 expf kx tω α= + − + with 2 2 4k kω = + ; the

two-soliton solution is generated by

1 2 1 21f E E AE E= + + + ,

where ( )exp 2i i i iE k x tεω α= − + with 22 1 4i i ik kω = + ,

2 2 4

1 2 1 2 1 22 2 4

1 2 1 2 1 2

( ) ( ) ( )

( ) ( ) ( )

k k k kA

k k k k

ω ωω ω

− − − − −= −+ − + − +

and 1ε = ± (describing propagation either to the left or to the right). In Figure 4 is shown a two-soliton solution of the Boussinesq equation.

Fig. 4 Perspective view of a 2-soliton solution of the Boussinesq equation, depicting a head-on collision

32

5.5 The two-dimensional Boussinesq equation The soliton solutions of the 2D Boussinesq equation follow exactly the same pattern as shown for the standard Boussinesq equation (in §5.4). The equation written in the form

23( ) 0tt xx xx xxxx yyu u u u u− + − − = ,

has the solution 2

22 lnu f

x

∂= −∂

, where ( , , )f x y t can be written

11 ef θ= + for the solitary-wave solution,

and 1 2 1 21 e e ef Aθ θ θ θ+= + + + for the 2-soliton solution,

where 1 2 1 2

1 2 1 2

( )( )

( )( )

p p q qA

p p q q

− −=+ +

. Here, we have introduced parameters ip and iq , so that

2 2( ) 3( )i i i i i i ip q x p q y tθ ω α= + + − − + ,

with 2

3 3 ii i

i

lk

kω = + ( )2 2;i i i i i ik p q l p q= + = − .

The 2-soliton solution is a quite general one of this type; however, a corresponding 3-soliton solution does not exist. The 2D Boussinesq equation is not completely integrable in the accepted sense (because it does not satisfy the 3-soliton test introduced by Hirota; see Hietarinta, 1987 and Johnson, 1996). But, as also happens in the case of the 2D KdV equation, we do have a 2-soliton resonant interaction, corresponding to a parameter choice that gives A = 0, as well as special – not general – N-soliton solutions (N > 2); two different soliton solutions of the 2D Boussinesq equation are shown in Figures 5 and 6.

33

Fig. 5 Head-on-oblique-collision solution of the 2D Boussinesq equation

Fig. 6 A resonant-wave solution of the Boussinesq equation

5.6 The Camassa-Holm equation The construction of the solution to the CH equation, written in the form

2 3 2t x x xxt x xx xxxxu u uu u u u uuω+ + − = + , 0ω > ,

34

is altogether much more complicated than any of the preceding methods. The description that we give here, based on Camassa & Holm (1993), Camassa et al. (1994) and Constantin (2001a), follows the implementation that can be found in Johnson (2003a). The soliton solutions, which may or may not be peakons, are generated by the choice

1

( , ) expN

i ii

ii

k kF X t X t α

λω=

= − − +

∑ ,

where 12

1 4i ik ωλ= + , 1

,04iλω

∈ −

; this is then used in the Marchenko equation

( , ; ) ( , ) ( , ; ) ( , ) d 0y

K y x t F y x t K y z t F z x t z∞

+ + + + =∫ ,

to give ( , ; )K y x t . We now construct d

( ; ) 2 ( , ; )d

Q y t K y y ty

= − , which provides the

coefficients in the ODE for ( ; )y tΦ :

14 0

4Q Q

ω ′′′ ′ ′Φ − + Φ − Φ =

,

the solution of which gives 2( ; )q y t = Φ . Finally, the solution that we require, ( ; )u y t , is expressed in parametric form (parameter y) as the solution of

12

qu q u u qω′′ ′ ′+ − = − with d

( ; )d

yq y t

x= .

The solitary-wave solution, obtained by using just one exponential term in F, can be written

2

2 2( 2 )sech

sech (2 ) tanh

cu

c

ω θθ ω θ

−=+

( 2 0c ω> > ),

with 00

0

cosh( )4ln

2 cosh( )

cx ct x

c

θ θθω θ θ

−− − = + − + , ( )0tanh 1 2 cθ ω= − ;

here, we have used the parameter θ rather than y. A corresponding 2-soliton solution is shown in Figure 7.

35

Fig. 7 Perspective view of a 2-soliton solution of the Camassa-Holm equation (T = 2ωt)

[One additional aspect of all these integrable equations is touched on in Exercises 16, 17: conservation laws associated with conserved densities.]

Lecture 6

6. Modulation of waves The analysis that leads to the Nonlinear Schrödinger (NLS) equation is certainly important in any general discussion of the theory of water waves, but it sits rather outside the type of problem that is of most interest in this course. Thus we will only briefly outline the method of derivation, and mention some of the results; more details can be found in the literature that we cite. The problem that we discuss retains arbitrary (linear) dispersion, so the underlying asymptotic limit will be: 0ε → , δ fixed; this is based on the original work of Hasimoto & Ono (1972). We seek a solution for which the initial wave-profile takes the form

( )( )exp i . .A x kx c cε + ,

where ‘c.c.’ denotes the complex conjugate and k is a general wave number; the governing equations are those that retain the parameter δ, so we use the set (1.10)-1.12) or (1.13)-(1.15). The appropriate ‘slow’ variables that we need here to describe the evolution of the wave are

px c tξ = − , ( )gx c tζ ε= − , 2tτ ε= ,

where ( )pc k and ( )gc k are the phase and group speeds, respectively. The asymptotic

expansion, at fixed δ for 0ε → , for the surface wave, takes the form

36

1

0 0

( , , ; ) ~ ( , ) . .n

n mnm

n m

A E c cη ξ ζ τ ε ε ζ τ∞ +

= =+∑ ∑ ,

where exp(i )E kξ= and 00 0A = ; all the various functions are expanded, following

this same pattern (with the z-dependence included, of course); we take 0b ≡ . The calculation is very lengthy and rather cumbersome, but fairly routine; by working as

far as the terms that arise at 2O( )ε , and balancing all the terms of the type mE , we find that

2 tanh( )p

kc

k

δδ

= and 12

[1 2 cosech(2 )]g pc c k kδ δ= + ,

for which the familiar identity ( )d

dg pc kck

ω ω= = holds. Finally, we obtain the

classical Nonlinear Schrödinger (NLS) equation for the leading term that describes the amplitude modulation of the surface wave:

2

201 01 01 012

d2i 0

dp pkc A kc A A A

kτ ζζ

ω β− − + = , (6.1)

where ( )2

2 2 42

11 9coth ( ) 13sech ( ) 2 tanh ( )

2p

kk k k

cβ δ δ δ= + − −

( ) ( )2 12 22 sech ( ) 1p g gc c k cδ− − + −

.

The relevance and application of this equation to our understanding of water waves is considerable but outside our main interest here); for more information, see Mei (1989), Johnson (1997). The extension to two dimensions, with a suitable weak dependence in the y-direction, is accomplished by allowing the initial profile to take the form

( )( , )exp i . .A x y kx c cε ε + ,

so the scale of the (slow) evolution in x and y is the same in this model. We follow exactly the same procedure as we did for the derivation of the standard NLS equation, but now we include Y yε= ; the resulting equation (cf. (6.1)) is

( )2 2 2

201 01 01 01 012 2 2

d2i

d 1p p p g YY

p g

kkc A kc A c c A A A

k c cτ ζζ

ω γβ − − − + + −

201 0k A fζγ+ = ,

37

where ( ) ( )22012

1 g YYp

c f f Ac

ζζζ

γ− + = − and 22 sech ( )p gc c kγ δ= + .

This coupled pair is usually called the Davey-Stewartson (DS) equations (see Davey & Stewartson, 1974), and this system is completely integrable for long waves ( 0δ → ); see Anker & Freeman (1978). Further, there is a correspondence between the KdV equation and the NLS equation for long waves (Johnson, 1997), and similarly between the 2D KdV equation and the DS equations (Freeman & Davey, 1975). 6.1 The inverse scattering transform for the NLS equation The Nonlinear Schrödinger equation, written in the form

2i 0t xxu u u u+ + = , (6.2)

possesses a rather different structure when compared with all our previous IST calculations: it requires a matrix formulation. (See Zakharov & Shabat, 1972 & 1974; also Drazin & Johnson, 1992). First, we introduce ( , , )F x z t , which here is a 2 2× matrix, and which satisfies the pair of equations

0 00

0 0x zl l

F Fm m

+ =

and i 0xx zz tF F Fα− − = ,

where l, m and α are arbitrary, real constants. Then the 2 2× function ( , ; )K x z t is a solution of the matrix Marchenko equation

( , ; ) ( , , ) ( , ; ) ( , , ) d 0x

K x z t F x z t K x y t F y z t y∞

+ + =∫ ;

it is convenient to set a b

Kc d

=

, and then we find that

( , ) ( , ; )u x t b x x t= and ( , ) ( , ; )u x t c x x t= ,

where the over-bar denotes the complex conjugate, gives a solution of the NLS equation

22 22i ( ) ( ) ( )( ) 0t xxl m u l m u l m l m u u

lmα − + + + − − = .

As an example, we outline the calculation that generates the solitary-wave solution

of the NLS equation. The choice for F is 1

2

0

0

fF

f

=

with

38

2 2 21 1 1 1exp ( ) i ( )f a mx lz l m tλ λ α = − + −

,

2 2 22 2 2 2exp ( ) i ( )f a lx mz m l tλ λ α = − + −

,

where ia and iλ (i = 1, 2) are arbitrary, real constants; solving for K (which is

altogether routine), and then extracting ( , ; )b x x t (and suitably redefining the various arbitrary constants), gives a solution of our NLS equation, (6.2), as

{ }12

exp i ( ) sech ( ) 2u a c x ct t a x ctω = − + − .

This solution – the solitary wave – expressed in terms of two arbitrary (real) constants a and c, represents an oscillatory wave packet that propagates at the speed c with a

maximum amplitude of a; the dispersion relation is ( )2 21 12 2

a cω = + . [Also see

Exercises 18,19.]

7. Variable depth Although our main interest hereafter will be on the important class of problems that generalise the basic description of flows (as given earlier) to include vorticity, we should mention another type of problem – one that is technically quite involved. This arises when we allow the wave to propagate over variable depth; in this situation, the leading-order governing equation is almost never of completely-integrable type. Indeed, to analyse it in any detail requires a careful discussion of many different elements (often, in this context, working to quite high order in the asymptotic expansion). As an example of what we encounter, we will outline the problem associated with the KdV equation in the presence of variable depth. 7.1 The Korteweg-de Vries equation for variable depth We consider (see §4.1) one-dimensional propagation, using the equations that have been scaled to remove the parameter δ; so we start with ( )T X z Xu uu Wu pε+ + = − ; { }( )T X z zW uW WW pε ε+ + = − ; 0X zu W+ = , (7.1)

with p η= & T XW uη ε η= + on 1z εη= + (7.2)

and ( )W ub X′= on ( )z b X= . (7.3)

One of the crucial questions in this problem is: on what scale should we allow the depth to vary? Let us write ( )b B Xα= , then we have, within a conventional asymptotic formulation, three distinctly different cases: o( )α ε= ; O( )α ε= ;

o( )ε α= ; alternatively, we could treat α as fixed (equivalently O(1)α = ) as 0ε → . This last possibility is by far the most difficult, so we will present some of the details

39

for the most mathematically interesting problem that is reasonably accessible: O( )α ε= . Thus we choose ( )b B Xε= and then introduce

1 ( )k Tξ ε χ−= − and Xχ ε= , (7.4)

where k accommodates the expected variable speed of the wave as it moves over variable depth. The characteristic variable, ξ, has been written to ensure that d d O(1)Xξ = i.e. the constant-depth case is recovered from the choice ( )k χ χ= :

T X Tξ χ ε= − = − . (Notice that this formulation uses a characteristic variable (for right-running waves) together with a large distance variable, rather than large time; cf. (4.1).) With this transformation to these far-field coordinates, equations (7.1)-(7.3) become [ ( ) ) ( )zu u k u u Wu k p pξ ξ χ ξ χε ε ε′ ′− + + + = − + ; (7.5)

{ }[ ( ) ]z zW u k W W WW pξ ξ χε ε ε′− + + + = − ; 0zk u u Wξ χε′ + + = , (7.6)

with p η= & ( )W u kξ ξ χη ε η εη′= − + + on 1z εη= + (7.7)

and ( )W uBε χ′= on ( )z B χ= . (7.8) We seek a solution of this set, following the standard form of asymptotic expansion, (4.5), and although, in principle, we could also expand k, that is unnecessary here; at leading order we find that

0 0p η= ; 0 0 ( )u kη χ′= ; ( )20 0( )W B z k ξη′= − (all for 1B z≤ ≤ ),

with 0

d( )

( )k

D

χ χχχ′

=′∫ where ( ) 1 ( ) ( 0)D Bχ χ= − > is the local depth.

Here, we have selected the positive square root to correspond to rightward propagation. At the next order, we obtain the KdV-type equation appropriate to this problem:

0 0 0 0 01 ( ) 3 1

2 02 3

DD D

DDχ ξ ξξξ

χη η η η η′+ + + = . (7.9)

This equation immediately recovers our familiar KdV equation, (4.6), when we take D = 1, but otherwise, for general depths, this equation is not completely integrable.

Often, the equation is rewritten with the change of variable 1 40( , )H Dξ χ η= , which

gives 7 4 1 21

2 3 03

H D HH D Hχ ξ ξξξ−+ + = ,

– more evidently a KdV-type equation. The depth-dependent factor in 1 40 D Hη −= is

usually called Green’s law. One final transformation is worthy of mention; with

40

20

ˆ ( , d )D H Dη ξ χ= ∫

and the choice 9 4( )D a bχ= + , for arbitrary constants a,b, the equation for H is the cKdV equation (which is completely integrable; see §§4.3, 5.3). Most of the work done on this equation, linked to a discussion of its rôle in wave propagation, has been for the case o( )α ε= ; in this situation, the leading term is the standard KdV equation, but with variable coefficients that depend on a new, third, variable. However, the interesting aspects are not restricted to the analysis of this equation; rather, it is the discussion of all the various wave-propagation phenomena that occur: primary wave, wave reflections and re-reflections. This problem is explored in Miles (1979), Knickerbocker & Newell (1980 & 1985), Johnson (1994).

Lecture 7

8. Weakly nonlinear, weakly dispersive waves with vorticity

We now turn to a discussion of a model that will more accurately represent another aspect of realistic flows. To this end, we consider the propagation of weakly nonlinear, (usually) long waves, of various types, in a flow that is moving according to some prescribed vorticity. (Because such a flow can be regarded as a model for a real flow, which can exhibit viscous and turbulent properties, this is often called a ‘shear’ flow.) The aim is to derive the leading-order approximation, based on a suitable asymptotic expansion, of the nonlinear surface wave in the presence of vorticity. This will then make accessible the possibility of describing the effects of a background flow on the propagation process (and the choice of this flow can model realistic, observed flows). The starting point is the set of governing equations, (1.10)-(1.12), which will be restricted so that they are appropriate to the discussion of plane waves propagating in the x-direction. Now we need to introduce the given background vorticity – and we take this to be O(1) relative to the scales previously defined – which we do in the form (for plane waves)

( ,0)uε ε⊥ =u is replaced by ( ( ) ,0)U z uε+ ,

where ( )U z is given. The equations therefore become ( )t x x z xu Uu U w uu wu pε′+ + + + = − ; (8.1)

[ ]2 ( )t x x z zw Uw uw ww pδ ε+ + + = − ; 0x zu w+ = , (8.2)

with p η= & t x xw U uη η ε η= + + on 1z εη= + (8.3)

41

and 0w = on 0z = , (8.4) where d dU U z′ = ; we shall consider only constant depth (so 0b ≡ ). Thus if, somewhere, the flow is an undisturbed parallel flow, ( ( ),0)U z , the vorticity is simply

( )U z′ . The set currently contains both our fundamental parameters, but we shall, for

some of the following calculations, invoke the scaling that removes 2δ in favour of ε; see (1.16). In this case, we obtain the equivalent set ( )T X X z Xu Uu U W uu Wu pε′+ + + + = − ; (8.5)

[ ]( )T X X z zW UW uW WW pε ε+ + + = − ; 0X zu W+ = , (8.6)

with p η= & T X XW U uη η ε η= + + on 1z εη= + (8.7)

and 0W = on 0z = . (8.8) 8.1 The Korteweg-de Vries equation with vorticity This problem is described by the set (8.5)-(8.8) and recast – as we did for the classical KdV equation (§4.1) – in suitable far-field variables, although in this case we do not know, a priori, the speed of propagation, c, of (linear) waves; thus we introduce the far-field variables

X cTξ = − , Tτ ε= .

Equations (8.5)-(8.8) then become

( ) ( )zU c u U W u uu Wu pξ τ ξ ξε′− + + + + = − ;

{ }( ) ( )z zU c W W uW WW pξ τ ξε ε− + + + = − ; 0zu Wξ + = ,

with p η= & ( ) ( )W U c uξ τ ξη ε η η= − + + on 1z εη= +

and 0W = on 0z = , and we seek an asymptotic solution exactly as before; see (4.5). At leading order, we obtain the problem described by

0 0 0( )U c u U W pξ ξ′− + = − ; 0 0zp = ; 0 0 0zu Wξ + = ,

with

0 0p η= & 0 0( )W U c ξη= − on 1z = ,

and 0 0W = on 0z = .

The appropriate solution of this set, defined for [0,1]z∈ , is

42

0 0p η= ; 0 2 0( )W U c I ξη= − ; [ ]0 0 2d

( )d

u U c Iz

η= − − ,

where [ ]2 2

0

d( )

( )

zz

I zU z c

′=

′ −∫

and then c is determined from

2(1) 1I = or [ ]

1

20

1( )

dz

U z c=

−∫ . (8.9)

(In the light of the notation developed below, we will write this in the shorthand

21 1I = .) This last result, (8.9), is the famous Burns condition (Burns, 1953), which,

for some choices of ( )U z , can lead to the presence of critical layers (i.e. where ( )U z c= for some (0,1)z∈ ), even for infinitesimally small waves; for more on these

ideas see Benney & Bergeron (1969) and Johnson (1986). [Exercise 20.] We will consider, here, only situations where critical layers do not appear in the linearised problem. The procedure at the next order follows the pattern laid down for the classical KdV equation (§4.1); this produces 31 0 41 0 0 1 02 3 0I I Jτ ξ ξξξη η η η− + + = (8.10)

where

[ ]

1

10

d(1)

( )n n n

zI I

U z c= =

−∫ and

[ ][ ] [ ]

21 1

1 2 20 0

( )d d d

( ) ( )z

U cJ Z z

U z c U Z c

ζ ζζ

−=

− −∫ ∫ ∫ .

(More details can be found in Freeman & Johnson (1970), which describes a development, and generalisation, of the seminal work presented in Benjamin (1962).) If we set aside the complications that arise when critical layers are present, the results represented by (8.10) are most encouraging. This is a KdV equation, with appropriate constant coefficients, which has, for example, solitary-wave and soliton solutions – and this holds for any background vorticity. Thus we can expect that these solutions will be relevant to wave propagation in (almost) any realistic one-dimensional flow of water. As a simple example, let us examine the special case of constant vorticity. We suppose that the original equations (expressed in X, T, u variables) have been written in the Galilean frame in which (0) 0U = ; indeed, this will be the physical frame if ( )U z models a ‘shear’ flow with a no-slip condition on the bed. Now we write ( )U z zγ= , for γ = constant, and then the speeds of small-amplitude waves are given (from (8.9)) by

214

2c γ γ = ± +

, (8.11)

43

and no critical layers exist for this choice; then, correspondingly, the KdV equation (8.10) becomes

( )2 20 0 0 0

14 3 1 0

3 cτ ξ ξξξγγ η γ η η η ± + + + + − =

. (8.12)

(We see immediately that the problem of zero vorticity, 0γ = , recovers our classical KdV equation, (4.6), for propagation to the right i.e. the upper sign.) It is of some interest, and possible relevance to more general theories of waves with vorticity, to investigate how the surface-wave profile is affected by the underlying shear flow, according to this model. To this end, we might consider two cases: γ ω= ∓ (= constant), where the upper sign corresponds to positive vorticity in much of the rigorous work that has be done in recent years. (This mismatch has arisen because here we have elected to use the (x, z) coordinate system, whereas the other work uses the (x, y) system, both being interpreted in the conventional (x, y, z) right-handed, rectangular Cartesian system, and then the two vorticities differ by a sign.) We should note, however, that here the choice of sign for the constant vorticity is immaterial: the underlying flow is then either to the left or the right – both generate the same mathematical problem – and then the crucial identification, in either case, is whether the surface wave is propagating upstream or downstream. To exemplify the essential character of the effect of constant vorticity, Figure 8 shows three solitary waves, all of the same amplitude, and all exact solutions of equation (8.12). The middle profile is for zero vorticity – the irrotational case 0γ = ; the outer (‘wider’) solitary wave is an example ( 1γ = ) of the profile obtained for upstream propagation (the sign of γ is immaterial), and the narrower profile is that obtained for downstream propagation. This observation – equivalently a longer wave upstream and shorter wave downstream – is precisely that made by Benjamin (1962). Similar observations apply to other solutions of this KdV equation, such as soliton interactions and cnoidal waves.

Fig. 8 The solitary wave of the KdV equation, for constant vorticity: middle profile is γ = 0;

outer (broader) profile is γ = 1, upstream propagation; inner (narrower) profile is γ = 1, downstream propagation.

8.2 The Boussinesq equation with vorticity We now turn to a discussion of a relatively new problem (for which the investigation is still ongoing). The derivation of the Boussinesq equation, §4.5, as we have seen, is considerably more involved than the corresponding problem for, say, the KdV equation. In the light of this, and also to make the details more transparent, we will examine the problem of a obtaining a Boussinesq equation appropriate for propagation

44

in the presence of constant vorticity ( ( )U z zγ= , for constant γ). The governing equations are therefore (8.5)-(8.8), with this choice of U, which give ( )T X X z Xu zu W uu Wu pγ γ ε+ + + + = − ; (8.13)

[ ]( )T X X z zW zW uW WW pε γ ε+ + + = − ; 0X zu W+ = , (8.14)

with p η= & (1 )T X XW uη γ εη η ε η= + + + on 1z εη= + (8.15)

and 0W = on 0z = . (8.16) The form of the asymptotic expansion is exactly as used throughout our discussions in this course (and specifically in the case for 0γ = ; §4.5). In this approach, as we have seen, it is necessary to expand the surface boundary conditions in Taylor expansions about z = 1, which we repeat here. Thus, at leading order, we find that, for

[0,1]z∈ , 0 0p η= , ( )0 0 0X T Xu η γη= − + , 0 0T Xu η= − , ( )0 0 0T XW zη γη= + , (8.17)

which gives 0 0 0 0TT XT XXη γη η+ − = , (8.18)

and ( , )0u X T satisfies this same equation (but related to ( , )0 X Tη as described in the

appropriate equations in (8.17)). At the next order, we find (for example) that

2 31 11 12 3

(1 ) (1 )T Xp Q z Q z η= − + − + ,

where 0 0( , ) T XQ X T η γη= + , and

{ } 31 11 1 0 02 3

( ) ( )T XXT XXX XX X XW Q Q u u z zγ η= + + + −

{ } 31 1 0 0 0 0 0( ) ( ) ( )TT XT X T X T Tu Q zη γη γ η η η η+ + + + − .

The equation for 1( , )X Tη is

( )2 2 2 21 1 11 1 1 0 0 0 0 0 0 03 2 2

( )TT XT XX XXXXXX

u u uη γη η η γ η γ η γ η+ − = − + + + + (8.19)

with

0 0 0 ( , )dTX

u X T Xγη η∞

′ ′= − + ∫ , (8.20)

45

assuming decay conditions ahead (X → ∞ ) of the wave. The equation for

0 1~η η εη+ , constructed from (8.18)-(8.20), becomes

( ) 2 2 2 21 10 0 03 2

(1 ) O( )TT XT XX XXXX XXu u uη γη η ε η γ ε γ η γ η ε + − = − + + + + +

(8.21)

where (if we need to use it) 0u satisfies (8.18), and with ( )0 0 0X T Xu η γη= − + ; this is

therefore a generalisation of equation (4.15) which, in that case, can be transformed into the completely integrable version of the Boussinesq equation. The new equation, (8.21), cannot, however, be transformed into anything equivalent to the classical Boussinesq equation, (4.18), by any transformation that corresponds to the Lagrangian form used in §4.5 i.e. X X→ using a transform linear in integrals of η and 0u , together with Hη → using all terms of degree two in η and 0u . (The

difficulties are readily seen by writing (8.21), first, in a frame moving at speed 12

γ

(which removes the term XTγη ), introducing 0h uη γ= − and noting that

21 10 02 2

(1 ) ~ ( )du u Tξ

γ γη η γ∞

+ − −∫ , with 12

X Tξ γ= − ; all this should be compared

with (4.15)-(4.17).) Nevertheless, the equation does admit (ignoring the 2O( )ε error

term) a 2sech solitary-wave solution:

( ) ( ){ }2 2 2 21~ sech 3 2 4 ( )

2 2

aa X cTη ε γ γ γ γ

+ + ± + −

,

where 212

( 4 )c γ γ= ± + (from (8.11)). This solution exhibits the same properties as

observed for the solitary-wave solution of the KdV equation, (8.12), for upstream/downstream propagation when 0γ ≠ . This is to be expected because the far-field approximation of (8.21), following either left- or right-going waves, recovers precisely the KdV equation (8.21), to leading order as 0ε → . Now the fact that we obtain the two variants of the KdV equation from the one Boussinesq equation, and each of these admits scaling transformations that convert them into any desired, standard version of the KdV equation for any γ (a property of the KdV equation over any shear) explains the difficulties mentioned above. The scaling transformation depends on c, in particular the direction of travel, and we cannot define a single transformation that will accommodate both directions of travel at the same time: either one or the other (as in the KdV equation), but not both. Thus we cannot transform our new Boussinesq-type equation into a standard Boussinesq equation. So we have a new equation, (8.21): Boussinesq with constant vorticity; we can expect, therefore, that this equation may describe new phenomena (but this is for future study).

46

8.3 The Camassa-Holm equation with vorticity We now examine the problem posed by considering the possible existence of a CH equation describing a class of water waves with some background vorticity. Although the problem for general vorticity can be formulated – this procedure is outlined in Johnson (2003b) – we will present the details for the case of constant vorticity (U zγ= ). The method follows that for the case of zero vorticity (§4.7), except that we must use

( ), ,x ct t w wζ ε τ ε ε ε= − = = ,

where c is given by the Burns condition

1

20

d1

( )

z

z cγ=

−∫ i.e. 214

2c γ γ = ± +

;

see §8.1 and equation (8.11). It turns out that, in the presence of an underlying vorticity, the equation for the horizontal component of the velocity in the flow at a specific depth (u ) must be replaced by

2 2 4ˆ ˆ ˆ( ) O( , )v u c uεβ ε δ= + + , (8.22)

for some constant β (which is zero for 0γ = i.e. for 1c = ± ). (It is more convenient, and simpler, to express the various constants in terms of c rather than γ.) The procedure is exactly that described in §4.7, with only fairly minor adjustments (in addition to the use of v above), resulting in a CH equation for v . With the choice

5 4 2

4 2 2 2( 2)

2( 1)(1 )

c c c

c c cβ + −=

+ + +,

which we note is zero for 1c = ± , and selecting, this time,

4 2

2 4 26 3

2(1 )( 1)

c c

c c cµ + +=

+ + +,

we obtain

{ }2 4 2 2ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ(1 )( ) ( 1) ((1 3 ) )x x xxxt xxtc v v c c vv c v vε εδ µ µ+ + + + + + − −

( )4 2

2 2 3 4ˆ ˆˆ ˆ ˆ ˆ2 2

6 32 O( , )

6(1 )x xx xxx

c cv v vv

cε δ ε εδ+ += + +

+. (8.23)

Then, with the scaling transformation

47

( )22

4 2

(1 )1ˆ ˆˆ ˆ ˆ(1 3 ) , 31

ccx c t v v

c c

µζ µµ

++= − →+ +

,

we obtain

{ }2 2 2ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ2 3 2t t

v v vv v v v vvζ ζ ζζ ζ ζζ ζζζκ ε εδ ε δ+ + − = +

(to this order) exactly as in (4.26), where

3 24 22 2

4 2

2 1(1 )

3 6 3

c cc

c c cκ

+ += + + + .

The depth at which the horizontal component of the velocity is defined for (8.22) (to produce v ) is now

12 10 8 6 4 2

0 2 4 2

2 16 33 31 32 3 3

6(1 )( 1)

c c c c c cz

c c c

+ + + + − −=+ + +

.

In conclusion, we have demonstrated that, just as for the corresponding KdV problem – but not the Boussinesq case – the inclusion of vorticity (albeit constant here) does not fundamentally affect the underlying propagation structure. We still obtain a standard (constant coefficient) variant of the CH equation. The details of this result, and how it impinges on the properties of this water-wave problem, are discussed in Johnson (2003b); here, we will simply note how the constant vorticity distorts the solitary wave, based on equation (8.23), for various γ. (The wave profile at the surface is recovered by using (8.22) in (4.23), but the dominant behaviour is given by ˆ~ vη .) In Figure 9 we show three examples – just as in Figure 8 – of exact solutions of (8.23); the previous observation (longer waves upstream, shorter waves downstream) is repeated here.

Fig. 9 The solitary wave of the CH equation, for constant vorticity: middle profile is γ = 0;

outer (broader) profile is γ = 1, upstream propagation; inner (narrower) profile is γ = 1, downstream propagation.

Comment: The corresponding analysis for the propagation of a modulated wave in the presence of vorticity (i.e. NLS over a shear flow) is a very considerable undertaking;

48

this will not be pursued in this course. A discussion of this problem can be found in Johnson (1976).

8.4 Ring waves with vorticity A very different type of problem which involves vorticity is provided by combining two types of geometry: a concentric wave expanding over a shear flow which moves in one direction. So we consider how a ring wave (§4.3) – and it may not be circular – propagates on the surface of water that is moving (in the x-direction, say) with some prescribed vorticity. In order to formulate this problem, we first take equations (1.10)-(1.12), with the parameter δ scaled out in favour of ε (via (1.16), retain the dependence on a second horizontal dimension (Y), and replace uε by U uε+ , for general U(z); thus we get ( )T X X Y z Xu Uu U W uu vu Wu pε′+ + + + + = − ; (8.24)

( )T X X Y z Yv Uv uv vv Wv pε+ + + + = − ; (8.25)

[ ]( )T X X Y z zW UW uW vW WW pε ε+ + + + = − ; 0X Y zu v W+ + = , (8.26)

with p η= & ( )T X X YW U u vη η ε η η= + + + on 1z εη= + (8.27)

and 0W = on 0z = . (8.28) We have also taken the bed to be horizontal (0b ≡ ). Our aim here is to provide an overview of this problem, highlighting some of the difficulties that we encounter, and producing some details for the case of constant vorticity. (This work is based on Johnson, 1990, although the specific choice of constant vorticity is not discussed in any detail there.) At the outset, we observe that the geometry of this problem is a complication (requiring a mix of rectangular Cartesian and polar representations). The full details, which will not be developed here, can be found in the reference cited above. The first stage is to transform to a plane-polar coordinate system that is moving (in the x-direction) at a constant speed c; this prescribes a moving frame, the speed of which may be assigned as we wish. Thus we transform according to cos , sinX cT r Y rθ θ= + = (8.29) with cos sin , sin cos ,u u v v u vθ θ θ θ→ − → + (8.30)

so that (u, v) now represents the horizontal velocity vector written in the polar system that is moving in the x-direction. Further, it is convenient to consider a large radius; it will then be possible to extend the analysis to a far field that accommodates (weak) nonlinearity and dispersion, and so correspond to our cKdV equation (§4.3), for example. The leading order, with this choice, will still generate the underlying linear problem (which is appropriate for weak nonlinearity in the near field). Thus we further transform according to

49

( ) , ( )rk T R rkξ θ ε θ= − = , (8.31) where ( )k θ is to be determined; the wave front represented by constantξ = , for given T, is a circle if ( ) constantk θ = , which recovers the concentric wave. Thus we incorporate (8.29) and (8.30), with (8.31), into equations (8.24)-(8.28); then we seek an asymptotic solution in the usual form (see (4.5)), to give, at leading order, 0 0 0 0( )( cos sin ) cosu U c k k u U W kpξ ξ ξθ θ θ′ ′− + − − + = − ;

0 0 0 0( )( cos sin ) sinv U c k k v U W k pξ ξ ξθ θ θ′ ′ ′− + − − − = − ;

0 0 0 00; 0z zp ku k v Wξ ξ′= + + = ,

with 0 0p η= and 0 0 0( )( cos sin )W U c k kξ ξη θ θ η′= − + − − on 1z =

and 0 0W = on 0z = ,

where d dk k θ′ = (and d dU U z′ = , as used earlier). It is altogether routine to find that a solution exists of this set (giving 0η arbitrary at

this order, which is to be determined at the next order; see below) provided that

( )[ ]

12 2

20

d1

1 { ( ) }( cos sin )

zk k

U z c k kθ θ′+ =

′− − −∫ , (8.32)

which is a generalisation of the Burns condition; cf. (8.9). At this stage, and in the context of our previous calculations, we choose to consider a constant vorticity flow:

( )U z zγ= (other choices are discussed in Johnson, 1990); in this case, (8.32) becomes

2 2

11 ( cos sin )

k k

k kγ θ θ′+ =

′+ −. (8.33)

Here, we have made the obvious choice: (1)c Uγ= = , and so we are in a frame that moves at the surface speed of the water. The first order, nonlinear ordinary differential equation, (8.33), has the general solution

2( ) cos 1 sink a a aθ θ γ θ= ± + − , (8.34)

which is real if the (real) parameter a satisfies 2 1a aγ≤ + , for a given γ. However, even when this is the case, this solution has the property that ( ) 0k θ = where

50

2tan

1

a

a aθ

γ=

+ −∓ ,

and this implies that r → ∞ on the wave front: a ring wave does not exist. Further, (8.34) does not recover the circular ring-wave solution, corresponding to

( ) constantk θ = , on a stationary flow i.e. when 0γ = ; we must conclude that (8.34) is not the solution that we seek. However, the equation also possesses a singular solution

21 12 4

( ) cos 1k θ γ θ γ= ± + , (8.35)

and here we elect to use the upper sign, so that 0k > (and we see that 1k = for 0γ = ). This solution accommodates all the properties that we seek and expect, most particularly the recovery of the circular ring-wave; the wave front, constantξ = , for a given T, is then expressed as

21 1

2 4

21 12 4

1

cos 1r

γ γ

γ θ γ

+ +=

+ +, (8.36)

which has been normalised so that 1r = at 0θ = . Three examples of the wave front, based on this description and including the circular case ( 0γ = ), are shown in Figure 10. The distorting effect of the underlying flow, which is moving from left to right, is quite evident – the wave front is an ellipse – involving downstream propagation on the front edge of the wave front ( 0θ = ), and upstream at the back (θ π= ± ). There is no critical level at any point below this ring wave (although more general vorticity distributions can exhibit this complication; see Johnson, 1990).

Fig. 10 Shape of the ring-wave wave-fronts (half the profile is plotted) for propagation over

constant vorticity. The semi-circular profile is for γ = 0; the middle curve: γ = 1; outer (largest) curve: γ = 2. For each profile, the region near 1 (θ = 0) is propagating

downstream, and at the other end (θ = π) is propagating upstream. The O( )ε problem generates the equation for 0η (with 1η now arbitrary at this

order), because we have chosen to construct the problem in an appropriate far field; cf. §4.3. This produces, after extensive manipulation, an equation of the form

321 0 0 0 4 0 0 5 0 0R

AAA A A

R R θ ξ ξξξη η η η η η+ + + + = , (8.37)

51

where each ( )i iA A θ= is an involved function containing many integrals over z,

although careful analysis shows that 3 0A = for all shear flows. In the case of a ring

wave over stationary water (so the wave front is a circle), equation (8.37) becomes

0 0 0 0 01 1

2 3 03R R ξ ξξξη η η η η+ + + = , (8.38)

the concentric or cylindrical KdV (cKdV) equation (which is completely integrable); see (4.11) and §5.3. Fortunately, the absence of the θ-derivative term in (8.37) means that this equation can be treated as a version of (8.38) under a scaling transformation (because θ now appears only as a parameter), but we must note that the equation is only completely integrable if 1 2 2A A = . In detail, with ( )U z zγ= , we find that

15 3

A = , and

21

1 22 cosA γ γµ θ= + + ;

( ) ( ) ( ) ( )( )( )

2 2 3 21 1 12 2 4

2 21 12 2

cos 1 cos cos 1 cos

cos 1 cosA

µ µ γ θ µ γ θ γ θ γ θ

γ µ θ γ γµ θ

+ + + + + + =+ + +

;

( )( )21 14 2 2

3 cos 3 cosA γ γ µ θ γ γµ θ= + + + + ,

where 214

1µ γ= + . (The check for the case 0γ = follows directly.) Then the scaling

transformation

1 32

ˆAξ ξ−= , ( )2 30 4 02 ˆ3A Aη η= ,

converts (8.37) into

ˆ ˆ ˆ ˆ0 0 0 0 01 1

ˆ ˆ ˆ ˆ ˆ3 03R R ξ ξξξαη η η η η+ + + = , (8.39)

where 1 2( ) A Aα θ = , and this is equal to 2 only for 0γ = , but otherwise this takes the

standard form, (8.38). Equation (8.39) is not completely integrable, for general γ, nor does the standard form of solitary-wave solution exist (based on the integral of the square of the Airy function, Ai), although a similarity solution – which is not physically realistic – does. This is not the place to begin an extensive discussion and analysis of this equation; suffice to note that this considerably more complicated flow problem – a ring wave over a shear flow – can be tackled by our techniques. The wave front is readily described (and evident from observations in the field), but we are left with an interesting mathematical problem posed by equation (8.39), for general ( )α θ .

52

Lecture 8

9. Two examples: periodic waves with vorticity; edge waves

We conclude this discussion of nonlinear water waves by examining two rather different types of problem. They both involve a small parameter, so we may invoke our techniques of asymptotic expansions, but they relate only loosely to our previous examples. They do, however, show how broad are the problems that are encompassed by the classical water-wave equations. First, we will examine an application of considerable current interest, from the rigorous viewpoint: the propagation of periodic waves in the presence of constant vorticity, and then a rather unusual mode of propagation: edge waves along a beach. Some additional information relating to this approach for these two problems can be found in Johnson (2012). 9.1 Periodic waves over constant vorticity In recent years, there have been some important developments that have confirmed the existence of periodic waves – not necessarily of small amplitude – with arbitrary vorticity; see, for example, Constantin & Strauss (2004 & 2007) and Constantin & Escher (2011). These wave profiles, and the underlying flow field, can exhibit some quite unusual and extreme properties, not least the appearance of stagnation points (where the speed of propagation of the wave is equal to the speed in the flow at some level below the wave i.e. a critical level; see §8.1). In addition, there has been some illuminating numerical work, for both constant and variable vorticity, by Ko & Strauss (2008a,b). This work has proved the existence of such flows – and Ko & Strauss provide some graphical examples of them – but there is no analytical detail of the structure of the solutions; this is a gap that an asymptotic (parameter) approach might fill. This is not the place to provide all the background information that underpins the formulation – this is available elsewhere – but we will briefly outline how we obtain the relevant equations. In the context of the work presented earlier, it is convenient to start from equations (1.10)-(1.12), but we allow propagation only in the x-direction (and there is no dependence on y). Further, we simplify the description by using just one scale length for the non-dimensionalisation, so we set 1ε δ= = . The waves that we shall describe are steady, moving at a constant speed c, so we introduce x ctξ = − , and then write ( , )U z u cξ = − . Thus for a wave without stagnation in the flow field, we have 0U u c= − < everywhere, so the wave is moving faster than any point in the body of the fluid, or any point of the surface. Stagnation occurs wherever 0U u c= − = i.e. the horizontal component of the velocity vector, in this frame, is zero. However, the usual formulation of the problem as adopted by many authors – and we will follow this approach – means that 0U = cannot be attained (although 0U → can still be examined). The equations (1.10)-(1.12) now become

zUU wU pξ ξ+ = − ; z zUw ww pξ + = − ; 0zU wξ + = ,

with p h= & w Uhξ= on ( )z h ξ=

53

and 0w = on z H= − , where the bottom is given by ( 0)z H= − < = constant and 0h = corresponds to the undisturbed free surface in this representation. Equivalently, the equations imply Bernoulli’s equation which, evaluated on the surface, is

2 2 2( ) constantU w h H Q+ + + = = on ( )z h ξ= , where Q is the conserved energy (often called the ‘total head’); we shall find that this is a useful equation in this discussion. The vorticity is constantzU wξ γ− = − = ,

where the sign (γ− ) here is a convenience (chosen to correspond to the sign used in much of the earlier work). It is appropriate, in the light of the transformation that we shall introduce shortly, also to define the stream function, ( , )zψ ξ : zU ψ= and w ξψ= −

and to write the total mass flux, in this moving frame, as

( ) ( )

0( ) d ( , ) d ( 0) constanth h

H H

u c z U z z pξ ξ

ξ− −

− = = − < =∫ ∫

and then to use 00 ( 0)pψ≤ ≤ > , corresponding to ( )h z Hξ ≥ ≥ − . At this stage, we

seek a solution in ( )H z h ξ− ≤ ≤ and ξ−∞ < < ∞ .

To proceed, we replace the unknown free surface by a known, fixed, boundary. The transformation used in much of the rigorous work is the Dubreil-Jacotin transformation (Dubreil-Jacotin, 1934), which uses the variables ξ and ψ (rather than ξ and z). We set

( , )D z Hξ ψ = + , and then with

(1 )( )z Dψ ψ∂ ∂ ≡ ∂ ∂

and ξ∂ ∂ replaced by ( )( )D Dξ ψξ ψ∂ ∂ − ∂ ∂

(so that 1U Dψ= and w D Dξ ψ= ), we find that

2 2 32D D D D D D D D Dψψ ψ ξξ ξ ψψ ψ ξ ψξ ψγ+ + − = ,

with 2 21 (2 ) 0D Q D Dψ ξ+ − + = on 0ψ =

54

and 0D = on 0pψ = .

The implementation of this approach shows that stagnation cannot be reached, because this corresponds to Dψ → ∞ . (This difficulty is avoided in the alternative descriptions

given by Ehrnström & Villari, 2008, and Wahlén, 2009.) A convenient reformulation, with an asymptotic approach in mind, is provided by the rescaling

D d ω= and Q qω= , where γ ω= ± ( 0ω > ) is the constant vorticity; this then produces

( )3 1 2 22d d d d d d d d dψψ ψ ψ ξ ψξ ψ ξξ ξ ψψω−= − −∓ (9.1)

with 3 2 2 1 21 ( 2 ) 0q d d dψ ξω ω− −− − + = on 0ψ = and 0d = on 0pψ = . (9.2)

(In the terminology of much of the work cited above, positive vorticity is associated with the upper sign.) This restatement of the problem allows, directly, the analysis of, for example, the case of large ω, although we will not follow that route here. Finally, we note that we require a solution (for ( , )d ξ ψ ) in

00 pψ≤ ≤ and π ξ π− ≤ ≤ ,

this latter being the choice for a periodic solution (and we take the crest to be at

0ξ = ); this is equivalent to regarding the original length scale, L, as the physical wavelength. The problem that we address here is one of the examples discussed in Johnson (2009), where more details can be found. We examine what is, essentially, a rather routine problem: the small-amplitude perturbation of a uniform state, but – as we shall see – this does provide a good example of expansion breakdown, rescaling and matching; see Chapter 3. We first need the uniform-flow solution, for positive vorticity (which produces the most interesting problem in this context), of equations (9.1)-(9.2); this is easily seen to be

0( ) 2 2d b b pψ ψ= − − − , (9.3)

with

( )3 202 2q b b b pω−= + − − , (9.4)

where we require (for a real solution) 02b p≥ . The arbitrary (real) constant, b, is

associated with the speed of the uniform flow on the bottom (say) because, from

1 2U D bψ ω ψ= = − − ,

we obtain 02b pω− − on the bottom.

55

Let ε measure the (non-dimensional) amplitude of the wave that perturbs the uniform state. The main aim here is to find an explicit representation of the structure of the flow field, and to examine the nature of near-stagnation, and so produce details not evident in the rigorous approach (although some of this can be seen in the numerical results). We seek a solution in the form

01

( , ; ) ~ ( ) ( , )nn

n

d d dξ ψ ε ψ ε ξ ψ∞

=+∑ (9.5)

where 0( )d ψ is the uniform state defined by (9.3); we consider 0ε → , at fixed

0, ,b p q and ω. In this exercise, we note that the asymptotic parameter does not appear

in the governing equations: we have introduced ε as the amplitude of the wave-solution

that we seek. The asymptotic sequence, { }nε , is the natural one to choose, because a

term 1ε , appearing in the equations and surface boundary condition by virtue of (9.5),

automatically generates a term 2ε , and so on. The problem that defines 1( , )d ξ ψ is

2 1 21 0 1 0 13( ) ( )d d d d dψψ ψ ξξω−′ ′− = − (9.6)

with

{ }3 2 20 1 0 0 1 0 12 2 2 ( ) 0qd d d d d d dψ ψω−′ ′ ′− + + = on 0ψ =

and

1 0d = on 0pψ = ,

where 0 0d( ) dd d ψ′ = . All higher-order terms each satisfy a similar partial differential

equation, with solutions that are readily constructed. Equation (9.6) possesses an exact solution which is periodic (period 2π ), with a peak at 0ξ = , and which satisfies the

bottom boundary condition (on 0pψ = ):

( )1 0cos 1

( , ) sinh 2 22

d b b pb

ξξ ψ ψψ ω

= − − − − , (9.7)

with

( )0 1

1tanh 2

bb b p

bω ω ω− − − = +

, (9.8)

which can be interpreted as defining b, for given 0p and ω, and then q is given by

(9.4). (Note that equation (9.8) has real solutions for b only if 00 pω κ< ≤ , where

7 009κ ≈ ⋅ , which corresponds to 02b p= .) Higher-order terms merely add small

corrections to the results just presented; indeed, these terms generate the expected higher-harmonics that are driven by cos ξ. The resulting asymptotic expansion is uniformly valid for π ξ π− ≤ ≤ and 00 pψ≤ ≤ , for any 0( 2 )b p− fixed as 0ε → ;

then the first two terms of a uniformly valid expansion give

56

( )0 0cos 1

~ 2 2 sinh 2 22

d b b p b b pb

ξψ ε ψψ ω

− − − + − − − − , (9.10)

subject to (9.8) and (9.4). Solutions exist (the rigorous approach tells us this) for 02b p→ ; this is the case that

we now examine. With this choice of parameters in the problem, we see now that the asymptotic expansion given in (9.10) is not uniformly valid – it breaks down – if

02b p→ , where 0pψ → (remember that 0[0, ]pψ ∈ ). That is, the asymptotic

ordering of the terms (O(1), O( ),ε etc.) no longer holds; this heralds the breakdown of the asymptotic expansion. The precise identification of this breakdown leads to a new scaling and a resulting reformulation of the original problem. The breakdown evident

in (9.10) occurs with the particular choice 202 O( )b p ε− = , and then where

20 O( )pψ ε− = , which corresponds to a region near the bottom boundary of the flow.

(Note: the higher-order terms in the asymptotic expansion are easily confirmed to be

no more singular than ( )02 2n

pε ψ− for each nndε .)

To proceed, we first define 2 202b p ε λ− = ( 0λ > ) and then from (9.10), for

0 O(1)pψ − = , we obtain

0 00

cos 1~ 2 2 sinh 2 2

2 2d p p

p

ξψ ελ ε ψψ ω

− − + − − , (9.11)

correct at O( )ε . As we have just observed, this asymptotic expansion is not uniformly

valid for 20 O( )pψ ε− = , and so we introduce 2

0pψ ε= − Ψ ( 0Ψ ≥ ); we then see

that this implies O( )d ε= , and hence we also introduce ( , ; )d dε ξ ε= Ψ . The problem

for d then becomes

( )2

3 2 22

with 0 on 0,

d d d d d d d d d

d

ξ ξ ξξ ξεωΨΨ Ψ Ψ Ψ Ψ ΨΨ

+ = − −

= Ψ =

(9.12)

the condition at the surface being accommodated through solution (9.11) to which d must be matched. We seek a solution of (9.12) in the form 0( , ; ) ~ ( , )d dξ ε ξΨ Ψ as 0ε → , which

gives directly

0 2 ( ) ( )d A Aξ ξ= Ψ + − ,

where ( ) 0A ξ ≥ is an arbitrary function. We now invoke the matching principle, and apply it to the two-term asymptotic expansion (9.11) and the one-term expansion

0~d d ; matching occurs only if

57

1( ) cosA ξ λ ξ

ω= − (so 1λ ω≥ ),

which gives the leading-order solution

( ) ( )21 2 1 2~ 2 cos cosd λ ω ξ λ ω ξ− −Ψ + − − − , (9.13)

for 0ε → and O(1)Ψ = . (It is fairly straightforward to confirm that (9.13) is the first term of an asymptotic expansion valid as 0Ψ → . Thus no further asymptotic (scaled) regions are necessary for the description of this problem.) The solution represented by (9.11) and (9.13) are appropriate to the parameter ω close to 0pκ (because b is close

to 02p ), and so we must have, in this case,

( )3 2 20 02 2 2 O( )q p pω ελ ε−= + − + , 0ε → .

It is instructive to examine the predictions implied by (9.13), which is the (asymptotic) solution for d valid near the bed (i.e. on 0pψ = or 0Ψ = ). For example,

we can see if this admits a stagnation point. By virtue of the Dubreil-Jacotin transformation, we obtain

( )21 2

0

1~ 2 cosU

D d dψ ψ

ω ε ω ε ω λ ω ξε

−

Ψ= = − = − Ψ + − , (9.14)

to leading order, which shows that indeed a stagnation point is approached on 0Ψ =

(the bed), and then only below the crest (0ξ = ), if 1λ ω= . An example of a solution with a stagnation point, based on our asymptotic results, is represented by the streamlines (Figure 11) and the associated behaviour of U on the vertical line below the crest (Figure 12).

Fig. 11 An example of the streamline pattern for small-amplitude waves over constant vorticity.

58

Fig. 12 The horizontal velocity component, U, in the flow below the crest

(corresponding to the solution depicted in Fig. 11) for 0 0 6389p = ⋅ .

9.2 Edge waves over a slowly varying depth Edge waves are a rather unusual phenomenon: they are waves that propagate along a sloping beach, with an amplitude that decays exponentially away from the beach; they were first reported by Stokes in 1846. A general introduction to the various theories of edge waves can be found in Johnson (2007), and the work described here is based on Johnson (2005). An example of a typical edge wave is shown in Figure 13. Our approach is prompted by the work of Constantin (2001b), where a particularly simple parametric representation of the run-up pattern on the beach was described (for the classical problem of a beach with constant slope). (This is an example of an exact solution of the classical water-wave problem.) We will employ, in this discussion, a variant of a parameter expansion called the method of multiple scales.

Fig. 13 An example of edge waves along a gently-shelving beach. The method of formulation here requires a different starting point as compared with all our previous analyses. We introduce a non-dimensionalisation based on just one scale length, so we use equations (1.6)-(1.8) with 1δ = . The edge waves, which we will assume in this model are simply travelling waves, propagate in the y-direction,

59

which is along the beach, y−∞ < < ∞ ; the ocean will exist from the edge of the beach (the boundary being described by the run-up pattern) out to infinity (x → ∞ ). The bottom topography, which includes the beach profile, is given by a suitable ( )b b x= . Thus we have, initially, the set

D

Dp

t⊥

⊥= −∇u,

D

D

w p

t z

∂= −∂

, 0∇ ⋅ =u , (9.15)

with

p h= & ( )h

w ht ⊥ ⊥

∂= + ⋅∇∂

u on ( , )z h t⊥= x (9.16)

and

d

d

bw u

x= on ( ).z b x= (9.17)

Note that, in this statement of the problem, the undisturbed surface 0h ≡ , is now at z = 0. Guided by the exact solution written down in Constantin (2001b), we introduce suitable scaled variables consistent with the appearance of the slope in that solution. So we express this in terms of the small parameter ε , the magnitude of the slope ′b ( )0 and, further, we assume that the depth varies slowly on this scale; thus we shall define the bottom by ( ) ( )z b x B X= = − , X x= ε , with ′ =B ( )0 1. (The change of sign is merely a convenience.) The appropriate choice of characteristic variable, describing propagation along the beach (in the y-direction), and a ‘slow’ variable used to measure the behaviour of the solution seawards, are

ξ ω ε= −ℓy t , θ ε α ε= ′ ′− z10

( ; )X XX

d , (9.18)

respectively, where ℓ (> 0) is a given wave number in the y-direction, and ω (= constant) is to be determined, as is the function α ε( ; )X . In order to formulate the problem of edge waves, we note that, in Constantin (2001b), the velocity components,

( , , )u v w , are proportional, correspondingly, to ( )cos sin , sin ,sin sinβ β β β β ,

where tanβ is the uniform slope of the bottom; similarly, p and h are proportional to sinβ . With this in mind, u, p and h are transformed (rescaled) according to

( , , ) ( , , )u v w u v w→ ε ε ; ( , ) ( , )p h p h→ ε ; (9.19) the resulting non-dimensional, scaled equations (from (9.15)-(9.17)), written in the variables (9.18), are

D

D

u

tp pX= − +α εθb g ; D

D

v

tp= −ℓ ξ ; ε D

D

w

tpz= − ; (9.20)

α ε εθ ξu v u wX z+ + + =ℓ 0, (9.21)

60

with p h= & w h uh vh uhX= − + + +ω α εξ θ ξℓ on z h= ε (9.22)

and w uB X= − ′( ) on z B X= − ( ) , (9.23)

where D

Dtu v u

Xw

z≡ − ∂

∂+ ∂

∂+ ∂

∂+ ∂

∂+ ∂

∂ω

ξα

θ ξε εℓ .

These equations, (9.20)-(9.23), provide the basis for our discussion of the problem of edge waves (along a straight beach), interpreted in the limit ε → 0. We seek an asymptotic solution in the usual form, (4.5), and we construct the problem at each order, imposing appropriate uniformity conditions: the behaviour of the solution as θ → ∞ , ξ → ∞ and X → ∞ (and/or 0X → ). These will be

necessary in order to determine completely the earlier terms in the expansions. (This technique of introducing various versions of essentially the same variable, but scaled differently (as with θ and X here), with uniformity conditions invoked, is the method of multiple scales.) In addition to the expansion of the dependent variables, we must also allow here

α ε α ε α( ; ) ~ ( ) ( )X X Xnn

n0

2

+=

∞∑ ,

where the term εα1( )X is omitted because it can be subsumed into any amplitude function (which, in general, depends on X); in this particular problem – and exceptionally – we do not also expand the constant ω (but this could be included). The leading-order problem from (9.20)-(9.23) then generates the nonlinear set

− + + = −ω α αξ θ ξ θu u u v u p0 0 0 0 0 0 0 0ℓ ; − + + = −ω αξ θ ξ ξv u v v v p0 0 0 0 0 0 0ℓ ℓ ;

p z0 0= and α θ ξ0 0 0 0u v+ =ℓ ,

with p h0 0= on z = 0; note that 0w is absent at leading order.

This set of shallow-water-type equations has the particular exact solution (selected to correspond to the (linearised) solution found by Stokes, and others)

p h A X A0 0 0

2

2 02 21

2= = −( ) cose eθ θξ

ωℓ

; (9.24)

u A0 0= − ℓω

ξθe sin ; v A0 0= ℓω

ξθe cos , (9.25)

for arbitrary A X0( ) and ω ; we have selected α 0 = −ℓ so that θ ε~ − ℓX (i.e.

eθ → 0 as X → +∞ ). This solution is then consistent with the initial surface profile

61

h A x y Ax x~ ( ) cos( )0

2

2 02 2

2ε

ωe e− −−ℓ ℓ

ℓℓ

for some A x0( )ε . At the next order, we obtain the set of equations

− + + + + = − +ω α αξ θ ξ ξ θu u u v u v u u u p pX X1 0 0 1 0 1 1 0 0 0 0 1 0( ) ( ) ( )ℓ ;

− + + + + = −ω αξ θ θ ξ ξv u v u v v v u v pX1 0 0 1 1 0 0 1 0 0 1( ) ( )ℓ ℓ ;

− + + = −ω αξ θ ξw u w v w pz0 0 0 0 0 0 1ℓ ; α θ ξ0 1 1 0 0 0u v u wX z+ + + =ℓ ,

with p h1 1= & w h u h v h0 0 0 0 0 0 0= − + +ω αξ θ ξℓ on z = 0

and w u B X0 0= − ′( ) on z B X= − ( ) . (The boundary conditions on the surface, z h= ε , are rewritten to all orders as evaluations on z = 0 by assuming the existence of Taylor expansions about z = 0, a procedure adopted in our earlier problems.) This set has an appropriate solution on z B X∈ −[ ( ), ]0 , although its form is quite complicated; see Johnson (2005). It turns out

that the asymptotic expansions are uniformly valid as θ → −∞ and ξ → ∞ only if

A B BA A0 0

2

02′ + ′ = ωℓ

. (9.26)

With this condition imposed, the solution for 1h can be written

hA

B

A

BA A1

2

203

32

3

203

20 0

2

8 84 2= + − ′

FHG

IKJ

ℓ ℓ ℓ

ωξ

ω ωξθ θ θe e ecos cos

+ ′ −FHG

IKJ

ℓ ℓ

42

2 0

3

203

20

2

ω ωθ θA

A

BAe e . (9.27)

The uniformity condition, (9.26), is solved to give

A XB X

X

B X

X

0

21

2( )

( )exp

( )= ′

′

RS|T|

UV|W|

zωℓ

d, (9.28)

and the non-uniformity that is evident in 0 1h hε+ , as B → 0 for general ω , is

discussed in Johnson (2005); it turns out that this imposes a restriction on the modes that can be accessed by this approach. To the order at which we have obtained the details, the surface wave is

62

h X A AA

B( , , ; ) ~ cos cosθ ξ ε ξ

ωε

ωξθ θ θ

0

2

2 02 2

2

203

3

2 8e e e− +

RS|T|

ℓ ℓ

+ − ′FHG

IKJ

+ ′ −FHG

IKJUV|W|

ℓ ℓ ℓ ℓ

84 2

42

2

3

203

20 0

22 0

3

203

20

2

ω ωξ

ω ωθ θ θ θA

BA A A

A

BAe e e ecos .

We take the shore (beach) to be described by B X X( ) ~ as X → 0 , and so (9.28) gives

A X k X0( ) ~ β , β ω= −FHGIKJ

1

21

2

ℓ, as X → 0, (9.29)

where k is a constant which is determined by the amplitude of the wave for some X > 0. If A X0( ) , and all its derivatives, exist as X → 0, then we require

β ω= −FHGIKJ

=1

21

2

ℓn , n = 0 1 2, , , ....,

which recovers the classical result for the modes of linear edge waves (conventionally obtained via the properties of the Laguerre equation). We see that the exponential decay as X (and x) → +∞ ensures the validity seawards, and is the property that shows that we have a trapped wave here. Our main interest in this brief discussion of this problem is the run-up pattern on the beach; this is given by the intersection of the surface wave with the bottom profile there i.e.

z B X h= − =( ) ε

and with B X X( ) ~ as X → 0, we will take this to be

− −x A A~ cos0

2

2 02 2

2e eθ θξ

ωℓ

, (9.30)

at leading order. Equation (9.30) therefore describes the run-up pattern at the shoreline, with A X0( ) given by (9.29) (for β = n ) and θ ε= − = −ℓ ℓX x. The result expressed by equation (9.30) has been generated by a multiple-scale technique, and so we should treat X, θ and ξ each as O( )1 and independent; however, for the purposes of producing graphical results, we must revert to the original coordinates. Then it is useful to consider a suitable normalised version of this equation:

12 1 2

012

2 1 2+ −+

=− − − −µ ξ µZ

nZn Z n Ze ecos

( ), (9.31)

where Z x= ℓ , µ ε= −k n nℓ

1 (for n = 1 2, , ... ) and the root Z = 0 has been eliminated. It is quite straightforward to show that solutions exist of this equation that

63

are continuous, bounded and periodic for µ µ≥ >n 0 (for suitable µn ). The solutions for µ µ< n comprise closed curves, spaced periodically, which coalesce for µ µ= n to form two near-cycloids that meet at their cusps; for µ µ> n , these curves separate to become a pair of curves that correspond to trochoids. These profiles are analogous to the cycloid and trochoid given in Constantin (2001b), although here we have a pair in each case, and either is an acceptable solution – pointing either towards the shore or seawards. An example of a cycloid-like profile (n = 2, 2 21 54µ = ⋅ ) is presented in

Figure 14 (this being one of a pair), and a pair of trochoid-like profiles ( 2n = , 25µ = ), together with closed-curve solutions which are not physically realisable, is

shown in Figure 15. Solutions for µ µ< n do not describe a physically realistic phenomenon.

Fig. 14 A cycloid-like run-up pattern

Fig. 15 A pair of possible solutions, together with closed-contour solutions.

This approach can be extended to include a long-wave current (Johnson, 2005) and, if a background vorticity is added (Johnson, 2008), just one of the pair of solutions mentioned above is selected. A typical edge wave, presented as a surface plot which depicts the run-up pattern, is shown in Figure 16.

64

Fig. 16 A three-dimensional plot of an edge wave.

************ ********************

65

Exercises 1. The hodograph transformation

Transform equations (2.7) by using the hodograph transformation ( , )x x u c= ,

( , )t t u c= , to show that

4 3uu cc cct ct t− = ,

wherever the Jacobian ( , )

0( , ) u c c ux t

J x t x tu c

∂= = − ≠∂

. (Note that the characteristic

variables for this linear, second order PDE are 2u c− and 2u c+ ; cf. (2.8), (2.9).) 2. Exponential decay of the solitary wave

Assume that ~ ea α ξη − and ~ ( )ez α ξφ ψ − (both as ξ → ∞ ), and hence use

equations (2.10)-(2.12) to show that

( ) cosh( )z A zψ α= and 2 tanh( )c

αδαδ

= ,

where α is a real, positive parameter. 3. Stokes’ highest wave This wave, in the limit of the highest possible wave, has a discontinuous derivative at its peak (although the profile is symmetric); it is defined by the requirement that the particle speed at the peak is equal to the speed of the wave i.e. u cξφ= = at the peak.

This wave can be modelled by satisfying some, but not all, the appropriate conditions; the simplest model takes the complete representation of the profile to be that as given

in Exercise 2: ea α ξη −≈ . We then impose the conditions that prescribe (0)η and (0)η′ , and use c as given in that Exercise. Show that 1 161c ≈ ⋅ .

An improvement to this model is to take

2e ea bα ξ α ξη − −≈ + ,

to use the same conditions as previously, but add the requirement to satisfy 23 ( 1)V c M= − ; show now that 1 290c ≈ ⋅ .

[Accurate numerical work on the complete set of equations has shown that, for the highest wave, 1 286c ≈ ⋅ .]

[ 2 2c a= , (0 ) 1 3 aη δ α+′ = − = − with 2 tanh( )c αδ αδ= ;

then 2 2( ) tanh( )c a b αδ αδ= + = , 2 1 ( 3)a b αδ+ = , with

( ) ( )2 2 21 2 12 3 4

3 1 (2 )a ab b c a b+ + = − + .]

66

4. Integral identities for the small-amplitude solitary wave Obtain approximations to the three integral identities relating T, V, I, C and M (given

in §2.2) in the case that the solitary wave is approximated by the 2sech profile. That is, take the surface wave to be

2 12

~ sech 3η ε ε ξ with ~u ξφ η= ,

the form of this being developed in the discussion on the Korteweg-de Vries equation (§4.1).

[ 4 3M I C ε≈ ≈ ≈ ; 4 (3 3 )V T ε ε≈ ≈ .] 5. Asymptotic expansions I Obtain the first two terms in an asymptotic expansion of the function

1

( ; ) 1 e xf x xx

εεε εε

−− = + − + +

, 0x ≥ , 0ε > ,

for x = O(1) (and away from x = 0) as 0ε → . Also find the leading-order terms (only)

in the expansions valid for (a) O( )x ε= ; (b) 1O( )x ε −= . Show, based on the terms obtained, that your expansions satisfy the matching principle. (You should decide what terms to retain in order to complete the matching process.)

[ ( )1~ 1f x xε −+ − ; x Xε= : 11

~ 1 e1

XfX

−− − + +

; x χ ε= : ( ) 1~ 1f χ −+ ]

6. Asymptotic expansions II Obtain the first three terms in an asymptotic expansion of the function

( ) 1 22 4( ; ) 1f x x xε ε ε−

= + + , 0x ≥ , 0ε > ,

for x = O(1) as 0ε → . Show that your expansion is not uniformly valid as x → ∞ . In the two further asymptotic expansions that are required, find the first two terms in each, and confirm that they satisfy the matching principle. (You should decide what terms to retain in order to complete the matching process.)

[ ( )2 2 431 12 8 2

~ 1f x x xε ε− + − ; 1 3x Xε −= : 2 3 412

~ 1 ( )f X Xε− + ;

1 6X ε χ−= : 4 1 2 1 2 4 3 212

~ (1 ) (1 )f χ ε χ χ− −+ − + ]

67

7. A model PDE Consider the wave-propagation model,

( )2tt xx xx

xxu u u uε− = + , 0ε > ,

which embodies both weak nonlinearity and weak dispersion. Seek a solution

0

( , ; ) ~ ( , )nn

n

u x t u x tε ε∞

=∑ as 0ε → ,

which satisfies ( ,0; ) ( )u x f xε = , ( ,0; ) ( )tu x f xε ′= − ; find 0u and 1u .

Show that this asymptotic solution (based on the two terms that you have found) is not uniformly valid as t → ∞ (for any reasonable f e.g. sufficiently smooth and on compact support). Introduce a suitable pair of far-field variables, and hence find the equation defining the first term in the asymptotic solution valid in the far field.

[ { }14

~ ( ) 2 ( ) ( ) ( )u f x t tF x t F x t F x tε ′− − − + − − + where 2F f f ′′= + ;

0( , ; ) ( , ; ) ~u x t U Uε ξ τ ε= where x tξ = − , tτ ε= ,

with ( )20 0 02 0U U Uτξ ξξ ξξ

+ + = .]

8. KdV for left-going waves Repeat the derivation described in §4.1, leading to the KdV equation for right-going waves (equation (4.6)), for waves that propagate to the left.

[ 10 0 0 03

2 3 0τ ζ ζζζη η η η− − = , where X Tζ = + .]

9. Higher-order correction to the KdV equation Continue the calculation described in §4.1 to find the equation that defines 1( , )η ξ τ . In

the case of a travelling-wave solution, where both 0( , )η ξ τ and 1( , )η ξ τ are functions

of ( )cξ τ− , show that the equation can be integrated by seeking a solution of the form

1 0 ( )F cη η ξ τ′= − (where the prime denotes the derivative with respect to ( )cξ τ− ).

[ 2 31 71 21 11 0 1 1 0 0 0 0 0 0 03 4 12 6 36

2 3( )τ ξ ξξξ ξ ξ ξξ ξξξ ξξξξξη η η η η η η η η η η+ + = + + + ,

where the KdV equation for 0η has been used to afford some simplification.]

68

10. From Boussinesq to KdV Introduce a suitable choice of far-field variables into equation (4.17) (or we could use (4.15), but here we start with the standard version of Boussinesq) and hence recover, at leading order, essentially the KdV equation for right going waves (equation (4.6)). Repeat this calculation for left-going waves; see Exercise 8.

[At leading order: ( )13

2 3 0τ ξ ξξξ ξη ηη η± − − = where X Tξ = ± , Tτ ε= .]

11. KdV 3-soliton solution

Show that the choice: 3

1

exp( )ii

F θ=

=∑ , with iθ as defined in §5.1, produces a solution

which can be written as 2

2( , ) 2 lnu x t A

x

∂= −∂

, where

33 3

1 1 1

1 i ij i j ij ii i i

A E A E E A E= < = > < = >

= + + +∑ ∑ ∏

with 30exp 2 ( ) 2i i i iE k x x k t = − −

, 2 2( ) ( )ij i j i jA k k k k= − + ,

and <> denotes that j is to be chosen cyclically with respect to i. 12. Phase shifts for KdV solitons Construct the asymptotic form of the solitary waves that appear as t → ±∞ in the 2-soliton solution of the KdV equation written as

[ ]23 4cosh(2 8 ) cosh(4 64 )

( , ) 123cosh( 28 ) cosh(3 36 )

x t x tu x t

x t x t

+ − + −= −− + −

,

which is a special version of the solution given in §5.1. First, for 16x tξ = − fixed as t → ±∞ , show that

( )2 12

~ 8sech 2 ln 3u ξ− ∓ ,

and, correspondingly for 4x tη = − fixed, that

( )2 12

~ 2sech ln 3u η− ± .

Hence deduce that the taller wave (remember that the wave of elevation is u− , u as

above) moves forward by a distance 12

ln 3x = , and the shorter back by ln 3x = ,

relative to where they would have been if moving at constant speed (i.e. as if no interaction had occurred).

69

13. KdV rational solution Show that the KdV equation

6 0t x xxxu uu u− + =

has a (rational) solution

( )3

23

246

12

x tu x

x t

−=+

;

however, this is not a particularly useful solution because it is singular along 3 12 0x t+ = .

[Seek a solution 2 3 1 36 ( )u t f xt− −= , and observe that 13

(ln )f F ′′= − for

3( ) 12F η η= + ( 1 3xtη −= .] 14. 2D KdV 2-soliton solution See §5.2; write 1 2F E E= + , where

2 2 3 3( , , , ) exp ( ) ( ) 4( )i i i i i i i iE x z y t k x l z k l y k l t α = − + + − + + +

,

and show that the solution can be expressed as

( )2

1 2 1 22ˆ ˆ ˆ ˆ( , , ) 2 ln 1u x t y E E AE E

x

∂= − + + +∂

,

where ˆ ( , , , )i iE E x x y t= and 1 2 1 2

1 2 1 2

( )( )

( )( )

k k l lA

k k l l

− −=+ +

.

15. Solitary-wave solution of the cKdV equation See §5.3; show that a solution for F is

1 3i i( , ; ) ( )A ( )A ( )dF x z t f st x s s z s

∞

−∞= + +∫ ,

where f is an arbitrary function (but appropriate to ensure the existence of the integral) and iA is the Airy function. The solitary wave is usually regarded as that solution

which corresponds to the choice: (.) (.)f kδ= , where k is a positive, real constant and δ is the Dirac delta function. Hence find the solution

22 3 1 3 2

i2( , ) 2(12 ) ln 1 A ( )du x t t kt s s

ηη

∞− −

∂ = − + ∂

∫ , 1 3(12 )x tη = .

70

16. KdV conservation laws Find the first three conserved densities of the KdV equation

6 0t x xxxu uu u− + = ,

and so deduce that

d constantu x∞

−∞=∫ , 2d constantu x

∞

−∞=∫ , ( )3 21

2d constantxu u x

∞

−∞+ =∫ ,

for solutions, u, that decay sufficiently rapidly at infinity. (To obtain these, integrate appropriate forms of the KdV equation over all x.) 17. cKdV conservation laws

Find the first three conserved densities of the cKdV equation

16 0

2t x xxxu u uu ut

+ − + = ,

and so deduce that

d constantu t x∞

−∞=∫ , 2d constanttu x

∞

−∞=∫ , ( )26 d constantxu t u t t x

∞

−∞+ =∫ ,

for solutions, u, that decay sufficiently rapidly at infinity. (To obtain these, integrate appropriate forms of the cKdV equation over all x.) 18. The Ma solitary-wave solution of NLS Show that the NLS equation

2i 0t xxu u u u+ + =

has the solution

( )2 2 ( cos i sin

( , ) exp i 1cosh 2 cos

m m nu x t a a t

n amx

θ θθ

+ = + +

,

for all real a and m, where 2 21n m= + and 22mna tθ = ; Ma (1979).

[Seek a solution 2exp i (1 i )u a a t f g = + +

, for ( , ), ( , )f x t g x t both real.]

71

19. NLS conservation laws Show that the NLS equation (as given in Exercise 18) has these first two conservation laws

( )2 412

d constantxu u x∞

−∞− =∫ , ( )23

2d constantxxx xuu u uu x

∞

−∞+ =∫ .

[Form xt x x xtu u u u+ for the first, and t xxx xxxtu u uu+ for the second.]

20. Burns condition

For the ‘shear’ profiles: (a) 0 1 0( ) ( )U z U U U z= + − ; (b) 21( ) (2 )U z U z z= − ;

(c) 1

1

, 1( )

, 0 ,

U d zU z

U z d z d

≤ ≤= ≤ <

where 0 1,U U are constants, determine whether critical levels exist. (Assume that a

critical level does exist, use the Burns condition, (8.9), and then show that there is no critical layer for (a) and (b), but that (c) does possess one.) For more discussion, see Johnson (1991). [(a) Only solutions are 1 0,c U c U> < , so no critical level; (b) with a critical level, then

1/ ( 1)c Uα = < satisfies 21

1 11 ln 2 ( 1)

2 1 1 1U

α α α αα α

+ −− = −− − −

, which has only one

solution: 0α < – no critical level; (c) find that 21 1( )c c U c dU− = − , which has three

solutions, one of which satisfies 10 c U< < : a critical level exists.]

************ ***********************

72

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