Hamiltonian long-wave approximations to the water-wave problem

ELSEVIER Wave Motion 19 (1994) 367-389

Hamiltonian long-wave approximations to the water-wave problem

Walter Craig a Mark D. Groves bq* a Department of Mathematics, Brown University, Providence, RI 02912, USA

b School of Mathematical Sciences, Bath University, Claverton Down, Bath, BA2 7AY, UK

Received 7 May 1993; Revised 11 January 1994

Abstract

This paper presents a Hamiltonian formulation of the water-wave problem in which the non-local Dirichlet- Neumann operator appears explicitly in the Hamiltonian. The principal long-wave approximations for water waves are derived by the systematic approximation of the Dirichlet-Neumann operator by a sequence of differential operators obtained from a convergent Taylor expansion of the Dirichlet-Neumann operator. A simple and satisfactory method of obtaining the classical two-dimensional approximations such as the shallow-water, Boussinesq and KdV equations emerges from the process. A straightforward transformation theory describes the relationship between the classical symplectic structure appearing in the water-wave problem and the various nonclassical symplectic structures that arise in long-wave approximations. The discussion extends to include three- dimensional approximations, including the Kp equation.

1. IntrodIlction

The exact mathematical model for inviscid, irrotational, free-surface flows (that is, the water-wave problem) involves a variant of the Dirichlet-Neumann operator for Laplace’s equation. The Dirichlet-Neumann operator is a non-local, bounded, linear operator that maps Dirichlet boundary data for a harmonic function to corresponding Neumann data. The well-known approximate theories for long waves involve no such non-local operator. This disparity between the exact and approximate theories is the subject of the present paper. The Dirichlet-Neumann operator is not often written down explicitly in discussions of water-wave theory, but, as this paper aims to show, it is central to an understanding of the relationship between the water-wave problem and its long-wave approximations.

The mathematical studies of long waves (waves whose wavelength is very much longer than the water depth) undertaken in the nineteenth century by Airy [ I], Stokes [2], Lord Rayleigh [ 31, Boussinesq [ 4-7 ] and Ko- rteweg and de Vries [8] amount to a classical singular perturbation scheme in which small parameters appropriate to the desired approximation are used to expand the equations of motion in a formal power series. An approximate dynamical system is obtained when the power series is truncated. When applied to the water-wave problem, the process results in the non-local, bounded Dir&let-Nuemann operator being replaced by a differential operator (cf. Ref. 19, p. 471). This replacement leads to formidable technical and conceptual difficulties when it comes to justifying the approximation mathematically. Nevertheless, the classical approximate models

l Corresponding author.

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368 W. Craig, M.D. Groves/ Wave Motion 19 (1994) 367-389

have been extensively and effectively used in a wide variety of theoretical and experimental contexts. In addition, the model equations themselves often enjoy remarkable mathematical features which are seemingly independent of the exact situations from which they emerge. The KdV equation [ 81 is a striking example. It successfully pre- dicts the properties of solitary waves established empirically by Russell [lo] and has a rich mathematical theory associated with it (solitons [ Ill, inverse scattering [ 12 ] and solitary-wave stability [ 131).

A mathematical contribution to this discussion has been given by Craig [ 141, who showed that the solution of the KdV equation with generic inital data remains close to the solution of the full water-wave equations with the same inital data for a time interval inversely proportional to the small parameter in the KdV approximation. This approach appears to be the only way of justifying the classical long-wave approximations in a satisfactory mathematical manner.

A significant development in water-wave theory was the discovery by Zakharov [ 151 that the problem has a Hamiltonian structure. This analogy with classical mechanics was pursued by Luke [ 161 and Whitham [ 171 and discussed in more detail by Broer [ 181, Miles [ 19,201 and Benjamin and Olver [21]. (There have also been recent contributions to this theory, most notably by Radder [22].) It had been found that many of the long- wave approximations to the water-wave problem also have Hamiltonian formulations. Researchers were therefore presented with a new question about the relationship between the exact and approximated water-wave problems: what is the connection between their Hamiltonian structures?

Before discussing the answer to this question, let us first review the elements of infinite-dimensional Hamilto- nian dynamics. A Hamiltonian evolutionary system is a system of partial differential equations of the form

vt = Jgradij(w), (1)

where v (t ) describes a path in a Hilbert space P equipped with inner product (. , .). The function fi : V C P + R, defined on a dense subdomain V of P, is called the Hamiltonian function, the gradient is taken with respect to the inner product on P and J is a skew-adjoint operator called the structure map which defines the Poisson bracket {. , .} = (grad (. ) , J grad (. ) ) . This set-up fixes a Poisson structure on P [ 23, Ch. 71. If the structure map J is invertible the Poisson structure is termed a symplectic structure. In this context J is called the cosymplectic operator and its inverse K is the symplectic operator [23, Ch. 71.

The water-wave problem falls into a class of Hamiltonian evolutionary systems in which 2, E P = (L2 (X) 1” for some X c I%“. The structure map J does not depend on v, the n components of v depend additionally on a position variable x E X and the Hamiltonian function has the form

ii= H, J X

for some Hamiltonian density function H. Typically H depends upon spatial derivatives of v, and V is chosen so that @(v 1 is well defined. Sharp analytic questions are out of the scope of the present paper and here V is conveniently taken to be the Schwartz class of vector functions. If J is invertible then the symplectic structure on P is defined by means of the symplectic 2-form

i2=; s dv= A K dv,

X

where K = J-‘. Under transformations w = f (v ) between Hilbert spaces (which in this context are called Poisson maps),

system (1) becomes

WI = JfgracL,~(w),

where Jf is the skew-adjoint operator

W. Craig, MD. Groves/ Wave Motion 19 (1994) 367-389 369

(%)J(%)’ and grad,,, R is the gradient of a with respect to w. System (2) is a Hamiltonian evolutionary system of the form ( 1 ), possibly with a different structure map. A transformation of P for which Jr = J is termed canonical. Such transformations play an important role in the classical theory of Hamiltonian systems (e.g. see Ref. 124, Ch. 91).

The most significant recent contribution to the understanding of the relationship between the Hamiltonian structures of the full and approximated water-wave problems was made by Benjamin [ 9, p. 471. Benjamin observed that a Hamiltonian evolutionary system of the form ( 1) may be sensibly approximated by keeping the phase space and Poisson structure fixed and replacing the Hamiltonian density function by an approximation HA (w ) . The original Hamiltonian evolutionary system is then approximated by

wt = Jgrad&(v).

Such an approximation is automatically canonical in the sense described above. Now consider a Hamiltonian evolutionary system in which H depends upon a small parameter E. The previous remarks indicate how to construct a sensible Hamiltonian perturbation theory for this system. One fixes the phase space and Poisson structure and describes the Hamiltonian as a power series in E. If H depends smoothly on E, one may expand it in a power series

H(v,E) = Ho(v) + EHI(W) + ~*Hz(w) + ...,

and a sequence of approximations to ( 1) is then given by

vt = Jgrad ( CiN,oc’Hj(V)

> , N = 0,1,2 ,... . (3)

The Hamiltonian versions of the classical long-wave approximations to the water-wave problem may be derived from the full problem in precisely this manner. Derivations along these lines have been given by Broer [ 18,251 and Broer et al. [26]. It is important to realise that no extra rigour is introduced into the derivations of the long- wave approximations by this method. The earlier remarks concerning the difficulty in justifying the long-wave approximations are still true.

An alternative Hamiltonian perturbation theory was introduced by Olver [27,28] in the context of the two- dimensional water-wave problem. Olver introduced (co)symplectic operators depending both on position in phase space and on the small parameter E. In Olver’s approach, both the Hamiltonian and the symplectic operator are expanded as a series in powers of E and there is a rule to determine which terms are to be retained in the various approximations. Other approaches to perturbation theory focus upon the expansion of an equation or a solution rather than the Hamiltonian. If the Hamiltonian is analytic in e then solutions of an appropriate initial-value problem are generally analytic too, so that the ultimate results are likely to be independent of the method used to derive them. On the other hand, even for analytic Hamiltonians, families of solutions identified by periodicity, quasiperiodicity, scattering behaviour or other phenomena need not depend analytically, or even smoothly on e. This observation is reflected in such diverse areas as Amol’d diffusion (e.g. Ref. [ 291)) the RAM theorem (e.g. Ref. [ 301) and the phenomenon of asymptotics beyond all orders (e.g. see Ref. [ 3 1 ] ). In these cases the primary focus should be on the expansion of the Hamiltonian, rather than the solution.

The purpose of the present paper is to clarify and expand upon the issues discussed above. A Hamiltonian formulation of the water-wave problem is introduced in which the non-local Dirichlet-Neumann operator appears explicitly in the Hamiltonian. One may then directly perform the classical step of approximating the non-local operator by a differential operator. This process is carried out by the use of a new Taylor expansion of the Dirichlet-Neumann operator in terms of differential operators, an expansion which is related to the Hadamard variational formula (e.g. see Ref. [32, Section 5.1 J) and is of independent interest. Hamiltonian perturbation theory is then applied to produce a sequence of approximate Hamiltonians. This approach has several advantages

370 W. Craig, M.D. Groves/Wave Motion 19 (1994) 367-389

over previous schemes. The essential steps in the long-wave approximations are made explicit and carried out in undisguised form, the process is simple and systematic and the computations involved are much simpler than those in any previously-reported technique. In addition, one may use the scheme to effortlessly derive higher- order long-wave approximations. It is hoped that the present technique will prove useful in a wider range of physical problems. It has already been used with success in an investigation of long water waves in a uniform horizontal channel of arbitrary cross-section [ 33-351 and further research is planned into long-wave models for layered and stratified fluids and systems with bottom topography.

The long-wave theory below commences in Section 3 with derivations of bidirectional long-wave equations, that is equations without predetermined directionality imposed by the scaling. One may then specialise to obtain unidirectional models such as the KdV equation which describe waves travelling in one direction only. Olver [ 231 has recently examined the process of unidirectionalisation of Hamiltonian long-wave models and has concluded that the process necessarily involves a change in symplectic structure. The derivation of Hamiltonian KdV-style equations has been addressed by Olver[27,28] and Menyuk and Chen [ 361, and is discussed in the present context in Section 5.

The approximating schemes are easily extended to three dimensions, which procedure is undertaken in Sections 4 and 6. A survey of the possible three-dimensional long-wave theories is constructed via this route. Two classes of wave motions are considered: waves that are equally long in each horizontal direction, and waves that are long in one horizontal direction and very long in the other. The latter length-scaling is employed in Section 6 to derive the Kadomtsev-Petviashvili (KP) equation [37], which is in some sense the three-dimensional counterpart of the KdV equation. A new, simple Hamiltonian formulation of the KP equation is thus obtained.

2. The Hamiltonian formulation of the water-wave problem

The topic in question is the ii-rotational motion of an incompressible fluid with a free surface in two or three dimensions. In its undisturbed state the fluid domain D,-, is bounded below by a flat rigid bottom y = -h and above by the undisturbed free surface

S-J = {(x,O,z): (x,z) E W2}.

This paper considers motions for which the fluid domain D,, is the region bounded below by y = -h and above by the free surface given by the graph

S = {(X,Y,Z) :y = rlL%z,tl].

In terms of the velocity potential 8 (x, y, z, t ) the problem is to solve Laplace’s equation

GpXX + Q)yy + Q)ZZ = 0 in D,,

with boundary conditions

(4)

- = 0 on y = -h, an

?jt = (1 + # + $) l/r* X Z an on S, (61

61 + +jVg]’ + g? = 0 on S, (7)

the last two of which are the nonlinear kinematic and pressure conditions at the free surface (e.g. see Refs. [ 38, Section 2651 and [ 39, p. 4331). For wave motion on all of D,, the domain So for the horizontal variables is R2, and one assumes that ]Vg( + 0 as x2 + z2 + oo. For spatially periodic motions the lateral boundary conditions are that both q and Q, inherit the periodicity, and So is restricted to a period cell.

W. Craig, M.D. Groves/ Wave Motion 19 (1994) 367-389 311

Let @ denote the evaluation of 9, at the free surface S. The functions q and 9, if sufficiently well behaved, completely determine the motion of the fluid. The variable q fixes the fluid domain D,, and @ provides boundary data on S for a well-posed elliptic boundary value problem of mixed type for p in D,. The variable (tf, 0 ) E V L (L2 (Se2 ) )2 is therefore the natural phase-space variable for the water-wave problem (cf. Ref. [ 151).

Let us now introduce the Dirichlet-Neumann operator G(q), which is defined as follows. Fix q and @, let Q be the (unique) solution of the boundary value problem

A~I = 0 in D,,

dp=Q, on S,

ap -=O any=-h, an together with the appropriate periodic or asymptotic conditions on 0, and define

The operator G(q) is non-negative and self-adjoint on a dense subdomain of L2 (So) [ 401. The Hamiltonian structure of (4)- (7) may now be deduced (cf. Refs. [ 19,2 1 ] ). The total energy of the wave

motion is

ii= J

;grl’ dx dz + J #7p12 dx dy dz = J [;gs2 -I- f@GhW] dx dz. (8) SO rr, s,

Taking the variational derivatives of a, one finds that

This system is a Hamiltonian evolutionary system of the form ( 1) with structure map

0 1

( 1 -10 ’

acting on the space {v = (q, 0 )T E 2, E P = (L2 (SO) )2}. The operator .I is invertible, so that the water-wave problem has a symplectic structure, defined by the symplectic 2-form

9= J

(dvr\ d@) dx dz.

SO

3. Long-wave approximations to the two-dimensional water-wave problem

The goal of this section is to derive, within a classical long-wave scaling regime, an expansion of the two- dimensional water-wave Hamiltonian (8) of the form

kv,e) =&+&+...,

giving rise to a sequence of Hamiltonian evolutionary systems which are approximations of the water-wave problem. The basis of the method to be employed is the following theorem, proved by Coifman and Meyer [41].

Theorem 1. There is a constant c = 0 (h) such that the operator G(q) is analytic in r,~ E C” in the ball of radius c about 1 = 0.


The Taylor expansion of G (q ) about zero takes the form

G(V) = gGj(tll, j=O

where Gj(q) is homogeneous of degree j in v. The terms Gj(q) may be computed systematically by the use of a recursion formula used by Craig and Sulem [ 421. The recursion formula is included in the Appendix for completeness. The first few terms in the series are

GO = Dtanh(hD),

Gi (q) = DqD - Dtanh(hD)qD tanh(hD),

Gz(q) = -i (D’q’Dtanh(hD) + Dtanh(hD)q2D2- 2Dtanh(hD)rIDtanh(hD)~Dtanh(hD)),

where D = -iafax. In the two-dimensional water-wave problem there are two dimensionless scaling parameters, namely (Y = a/h,

and /3 = (h/e ) 2, where a is a typical wave amplitude and .! is a typical wavelength. One implements a long-wave scaling regime by specifying the relative sizes of the parameters. A detailed discussion of the standard choices has been given by Ursell [ 431.

3.1. The shallow-water scaling regime

This parameter regime describes wave motions where (Y = 0 ( 1 ), while B = p2 is taken to be a small parameter. The regime is introduced by the transformation to stretched variables

XI = W, t1 = M,

which induces the transformation

Pa D=pDIsaJcl.

The terms in the expansion of the Dirichlet-Neumann operator become

Go = pD1 tanh(phDI 1, Gl(q) = P~UQDI - DI tanh(~hDI)1DItanh(~hDI)),

and so on. One determines the dependence of g upon the small parameter by expanding

tanh(phD1) = phD1 - f(ph)‘Df + ...,

and using (8) to find that

@LW = J

[;gq2 + ;@ (p’Dl(h + tt)D~ - +p4D:(h + t113D: + . ..)I dx. (91

SO

A sequence of approximate Hamiltonians is constructed by the retention of terms up to O(p2” 1, n = 1,2,. . . in the integrand of (9). In carrying out this process one is assisted by the fact that the recursion relation implies that @Gj(q)@ is at least of order j + 1.

In the first approximation one evaluates @ to 0 (p2 1, so that

ii= J

[;gq2 + f(tt + Wp2@;,] dx + O(P’).

SO


Returning to unscaled variables, which is equivalent to setting ,u = 1, one may define the approximate Hamilto- nian

Hsw = J [fso’ + ;ol + we] d-c SO

so that the approximate equations of motion are

SHsw &- so - - = -((q + hmc),, (10)

(11)

One may write Eqs. ( 10) and ( 11) in a more revealing form by differentiating ( 11) with respect to x and writing u = OX. They become

fit = -[(tl + h)ulx, (12)

ut = - [gq + 3u2],, (13)

which are the classical shallow-water equations. The device of writing u = 0, will be discussed in more detail in Section 5.1.

In the second approximation the Hamiltonian is evaluated to 0 (~~1. One finds that

SO

where

The second approximations to the equations of motion are, in unscaled variables,

sR qt=m= -((tl + hl@P,l, - 4((t1+ h)2@XX)xx,

sfi &z-i= arl

-&V- $& + f&(tl+ h12.

One may again write u = QX to transform this system into

tit + ((h + tl)u)x + +w + ?)‘UxLx = 0,

ut + uux + gqx - [)(s + h)*zq = 0.

With further work one may derive higher-order approximate equations in a straightforward manner.

(141

(15)

3.2. The Boussinesq scaling regime

In this parameter regime one focuses on long waves which have a smalI amplitude compared to the depth of the water. That is, one chooses a = /I = p2 to be the small parameter. The regime is instituted by the introduction of stretched independent variables

XI = PX, t1 = Pt,

374 W. Craig, M.D. Groves/Wave Motion 19 (1994) 367-389

and scaled dependent variables

1 rtl = p (16)

0, = p. (17)

The Hamiltonian depends on fi through the induced transformation D = pDI as before, but now there is an additional dependence of @ on p through the small-amplitude scaling ( 16)) ( 17).

In this small-amplitude, long-wave regime one constructs a sequence of approximate Hamiltonians by expanding the Dirichlet-Neumann operator in the integrand of (8) and retaining terms up to O(p’“), n = 2,3,. . . . In carrying out this process one is assisted by the fact that Gj is homogeneous of degree j in r7 and of degree 2 in 0, so that 01 Gj (VI )@I is at least of order ~~+~j.

In the first approximation to the Hamiltonian one terms up to 0 (p4 ) are retained. All but the first term GO (~1 in the Taylor series for G( q ) may therefore be neglected, so that

The next step is to expand Go = pDl tanh (phD, ) and retain terms up to 0 (p4 ), so that the Hamiltonian becomes

The approximate equations of motion are, in unscaled variables,

Writing u = <p, as before, one finds that these equations reduce to the factored wave equation

t/t + hu, = 0, (18)

Uf + gq, = 0. (19)

The second approximation is obtained in a similar fashion. The first two terms in the expansion of G (q ) must now be retained, whereupon

= J

[isp’tl: + fp@l(D1 tanh(hDi) + DirllDi - Drtanh(hDr)‘Irtanh(hDr)Di)@i] dx + O@s).

SO

The next step is to expand pD1 tanh (,uhD, ) and retain terms up to O( cl”), so that the Hamiltonian becomes


The approximate equations of motion are, in unscaled variables,

(20)

SRB 91=~= -((q + h)@c)* - @3~xxxx, (21)

(22)

and, writing u = @,, one arrives at the more familiar system

tlr + ((h + ?)U)X + +h3uXXX = 0, (23)

Ut + UUX + gtjx = 0, (24)

which constitutes a properly Hamiltonian version of the Boussinesq equations. This version differs from the usual Boussinesq system as given, for example, in Ref. [ 391, but is the natural system that arises from the Hamiltonian perturbation theory. It has been shown to be completely integrable by Kaup [44] (see also Ref. [45] ).

One may easily proceed to higher orders of approximation. The third approximation to the Hamiltonian, for instance, is obtained by retaining terms up to 0 (p8 ). One finds that

ii = fi- + O(/?O),

where

The approximate equations of motion are given by

which in the usual notation are

rtt = - [(1 + h)ulx - fh3uxx* - h2[wxl*x - #z5Ux*xxx, (25)

Ut = -gqx - uux + +;]Jz? (26)

Steady wave profiles of a simplified version of this long-wave system have been studied by Zufiria [ 4_6]. The fourth-order approximation is not so well studied. It consists of the choice of Hamiltonian H = & +

O(p’*), where

376 W. Craig, M.D. Groves I Wave Motion 19 (1994) 367-389

In the usual notation the approximate equations of motion are

‘It = -[(tt + hhlx- fh'uxxx- h2[quxlxx- &h5uxxxxx

-373 I7 h’uxxxxxxx - ~h4biwxxl,, - bpxlxxxx - h[ft2u1xx,

ut = -gtlx - uux + fb:lxh2 + ~bwxxxlx + h[tp:lx.

(27)

(28)

One may proceed to higher-order approximations in a straightforward manner.

4. Long-wave approximations to the three-dimensional water-wave problem

It is not difftcult to extend the method presented in Section 3 to three dimensions. There is no analogue of Theorem 1 for the Dirichlet-Neumann operator in three dimensions, but there is a well-defined formal expansion

G(V) N eG,(T), j=O

where Gj (q ) is homogeneous of degree j in q (see the Appendix). In terms of the operator

D = ; (&&)‘,

the first few terms in the series are

GO = (DI tanh(h(DI), Go = D. tlD - GorlGo,

G2b) = -f (IDl’tlGo+ GottJD12--GorlGdlGo),

where

Notice that all the terms in the expansion are analytic in D. In the three-dimensional water-wave problem there are three dimensionless scaling parameters, namely o =

a/h, px = (h/e,)’ and PI = (h/t,) 2, where 4 is a typical wave amplitude and e,, e, are typical length scales in the x and z directions. Two kinds of long-wave scalings are possible: ones which are isotropic in the horizontal variables x and z and ones which have different length scales in the x and z directions.

The shallow-water scaling regime in the isotropic setting describes wave motions in which o = O( 1) and /3X = jIZ = p2. The regime is introduced by the transformation to stretched variables

XI = /lx, ZI = ,uz, t1 = pt.

One proceeds by expanding the Dirichlet-Neumann operator in powers of p2 as before. In the first and second approximations one writes the Hamiltonian as

ii= J

[$gq’ + fp2(tl + hl W:, + @;J] dx dz + W41,

SO


and

The lirst of these Hamiltonians gives the isotropic shallow-water equations. The second gives a system analogous to ( 14), ( 14) which includes dispersive corrections to the shallow-water equations.

The isotropic Boussinesq regime includes the additional expectation of small amplitude motions. One chooses LI = /IX = pz = p*, implemented by the introduction of stretched independent variables

Xl = /lx, zi = jlz, t1 = /Lt


vi = 1, P2

@, = +.

In the first and second approximations the Hamiltonian is written as

2 = fid + 0(/P),

where

i7 = R + 0(/P),

where

The first approximations to the equations of motion are therefore

s& It-&g= -hQixx - h@D,,, Gj = -!$ = mg,,,

the factored linear wave equation, and the second approximations are

& &= J@ - -[(tl+ hP&lx- [(q + h)@,],- bh3A2@,

# sii,

j=-6tl= -7 x-2 z-m ‘uj2 ‘fD2

a three-dimensional analogue of the Boussinesq system. A common nonisotropic spatial scaling regime considers wave motions which are long in one horizontal di-

rection and very long in the other. One chooses a = 0 ( 1 ), /3x’ = Bz = p4 by the introduction of stretched independent variables

x1 = W, zi = $z, t1 = pt.

378 W. Craig, MD. Groves/ Wave Motion 19 (1994) 367-389

The Hamiltonian is expanded in the usual fashion. In the first and second approximations it is given by

R= J

[fgl’ + jp2(rl + W:,] dx dz + 0(p4)

SO

and

Perhaps more interesting is the Boussinesq scaling regime in this nonisotropic setting, corresponding to the choice CY~ = j?: = pZ = p 4. One introduces stretched independent variables

X1 = /Ax, zi = $z, tl = pt


In the first and second approximations the Hamiltonian is written as

E = fif + oo&,

where

fif = J [fe4v: + +‘h@:x,] dx dz SO

and

ii = R* + O($),

The first approximations to the equations of motion are therefore

the factored wave equation in one spatial dimension, and the second approximations are

dH* q*=sQ>‘= -[(v + hP,l,-h@zz- fh3@xxxx,

(29)

(301

(31)

a precursor to the KP equation (Section 6).

W. Craig, M.D. Groves/Wave Motion 19 (1994) 367-389 379

5. The KdV equation

The KdV equation is usually derived from the Boussinesq equations (23 ), (24) by the restriction to unidirectional wave motion [ 81. It has a Hamiltonian structure in terms of a nonclassical symplectic structure [ 471. This difference in symplectic structure makes the relationship between the full water-wave problem, the Boussinesq system and the KdV equation a more complex issue. To describe this relationship from the present standpoint let us start with an alternative Hamiltonian formulation of the two-dimensional water-wave problem.

5.1. An alternative Hamiltonian formulation of the two-dimensional water-wave problem

This alternative formulation of the water-wave problem, proposed by Benjamin 19, Section 6.21, is achieved by the change of dependent variables (q, 0) + (q, u), where u = OX. A description of the Hamiltonian in the new variables is provided by the following proposition.

Proposition 2. Under the hypotheses specified in Theorem 1, there is an operator K(q) such that

G(V) = DK(qP.

The operator K is positive, bounded and self-aa’joint on a dense subdomain of L2 (So ).

In the new variables the Hamiltonian is therefore

i?= J [;gtt= + fuK(rt)u] dx. (32)

s,

According to the transformation theory presented in Section 1, the water-wave problem in the new variables is

(33)

(34)

This system is a Hamiltonian evolutionary system of the form ( 1) in which 21 is the dependent variable (q, u)’ and J is the skew-adjoint structure map

(35)

It should be noted that the system has two elementary Casimirs

ij= qdx, J

ti= udx, J

so s,

that is JB grad ti and JB grad li are both identically zero. The structure map JB is therefore not invertible, which is to be expected because the transformation (Q @ ) + (q, u) is singular and .fB inherits this degeneracy. One may restrict the phase space D to the subset

{(?/,u) E D: i = li = 0).


In this restricted phase space the degeneracy is removed. One may invert the structure map JB and define a symplectic structure. In the new variables the symplectic 2-form is

%I= J

(dqAD,-‘) du dx,

SO

where 0;' is the inverse of the operator a/ax. A long-wave approximating scheme of the usual kind may easily be constructed for this alternative Hamiltonian

formulation of the water-wave problem. One begins by expanding the Dirichlet-Neumann operator G (1) as described in Section 3. Because

K(q) = D-'G(q)D-',

the operator K(q) may be expanded as

K(q) = 24(S), j=O

where Kj (q) is homogeneous of degree j in q and is given explicitly by the formula

Kj(q) = D-'Gj(q)D-'.

In the shallow-water scaling regime this expansion of K (q ) may be used to expand the Hamiltonian (32) as

fi(% u) = fio(tl) + $R(rl,u) + P4%(rl,U) + ***

by the method explained in Section 3.1. A sequence of approximate Hamiltonians and hence approximate equations of motion may now be constructed in the usual fashion. Here the first approximation will give Eqs. ( 12) and ( 13) (the shallow-water equations) and the second approximation will give Eqs. ( 14) and ( 15).

The Boussinesq scaling regime is implemented by the introduction of stretched independent variables

XI = W, t1 = put


1 1 ?I = y’l, U] = TU.

P

The first approximation will then give Eqs. (18) and (19) (the factored wave equation) and the second approximation will give Eqs. (23) and (24) (the Boussinesq system). The third approximation will give Eqs. (25 ) and (26) and the fourth Eqs. (27) and (28).

The above discussion demonstrates the fact that all the approximations derived in Section 3 also have the alternative Hamiltonian formulation. In this discussion the coordinate transformation (Q 0 ) + (q, u) was applied to the exact water-wave problem. It could, however, have been applied to any approximate theory. For instance, the Boussinesq system (20)-(22), written with the original symplectic structure, may be transformed, via this coordinate change, into a Hamiltonian system with Hamiltonian

i&j = J [fgp4d + f(P2?l + NP4d - g3P6&,] dx (36)

SO

and the alternative symplectic structure. The Boussinesq system will then be represented in the form of Eqs. (23) and (24). In Section 3 the device of writing u = @, was employed to write the systems in more recognisable forms. This device corresponds precisely to changing to the alternative Hamiltonian structure via the coordinate transformation (q, @ ) + (v, u).


5.2. A derivation of the KdV equation

The starting point in the derivation of the KdV equation is the Boussinesq system (23), (24) with the Hamil- tonian formulation (33), (34), in which 2 = & + 0 (p* ) and Es is given by (36). To simplify the working one introduces the nondimensionalisation

w (X,Y,Z) = $ (xt,Yt, ztl, t=t’ ; ) 0 u’

u=-, lm

which is equivalent to setting g = h = 1 in (23), (24), (36 ). Here a prime denotes a dimensional variable. The first step in the derivation of the KdV equation from the Boussinesq system is to transform to a frame

of reference which is stationary with respect to the right-moving (or alternatively the left-moving) wave front associated with the longest waves. In nondimensional variables the speed of translation of this wave front is unity. One carries out the transformation by subtracting the conserved quantity impulse

II = J url dx (37)

so

[ 9, Sections 1.2 and 1.31 from the Hamiltonian. The process is equivalent to the introduction of new independent variables

x2 = xi - t1, t2 = $t,

(cf. Ref. [ 9, p. 10, Eq. (2.6) ] 1. In the new frame of reference the Boussinesq system is

where

To focus upon right-moving evolution one makes the further change of variables

r = $(tl + u),

s= f(rl-u),

with corresponding scaled variables

n = -r, j2 51 = -+s. According to the transformation theory presented in Section 1, the system (38)-(40) transforms to

(38)

(39)

(40)

(41)

(42)

(43)

(44)

where

382 W. Craig, M.D. Groves / Wave Motion I9 (1994) 367-389

This system is a Hamiltonian evolutionary equation of the form ( 1) in which u is the dependent variable (r, s)~ and J is the skew-adjoint structure map

The system has two elementary Casimirs

and

s= sdx. ^ J

SO

The degeneracy may again be removed by restricting phase space to the subset

{(r,s)ED:i=i=O}.

One may now invert the structure map J mv and define the symplectic 2-form

&dv= J (drAD,-’ dr- d.sr\D,-’ d.r) d.~

so

on the restricted phase space. When written out explicitly in unscaled variables, Eqs. (43 ), (44) are

The change of variables (Q U) --t (r, s) therefore reduces the Boussinesq system to the above pair of coupled KdV equations (cf. Ref. [ 481). The crucial step in the derivation of the classical KdV equation is to concentrate on the region of phase space in which these equations decouple.

Let us therefore concentrate on the region of the phase space that corresponds to a predominantly right-moving evolution, that is the region where q is close to u in the H’ (So) norm. The Hamiltonians (40), (45) are defined in all of phase space V. Let A be a ,u2-neighbourhood of q = u in the sense that

A = {(rl,~) E ZJ :II tl- u IIH~(So)< ~‘1 = {(r,s) E 2, :I1 s IIH~tSOj< ~P’I,

and restrict the phase space to A. All but the second and third terms in (45) are then O(p’) and the Hamiltonian therefore reduces to

fi-z = fixdv +0(/P),


Returning to unscaled variables, one may write the system explicitly as

(47)

Here (46) is the classical KdV equation in the variable r and (47) is an equation which says that in the present approximation there is no change in s.

Some remarks concerning the validity of the KdV approximation are now called for. One may pose a mathematical problem by imposing the long-wave, small-amplitude scaling appropriate for the Boussinesq system, specifying initial data q (x, 0), u (x, 0) and allowing it to evolve according to the water-wave system (35)- (37). If one chooses initial data that lie in A then the KdV approximation will describe the subsequent evolution for as long as the solution remains in A (van Groesen and Pudjaprasetya [49, Section 4.11 have shown that there are solutions that remain in A for a finite time). It is important to realise that there is no a priori guarantee that solutions remain in A as time proceeds. A lower bound on the length of the time during which solutions evolve according to the KdV approximation has been obtained by Craig [ 141, but this bound is probably not optimal. A more precise estimate of the time of validity of the KdV equation would be of much interest.

5.3. Higher-order KdV equations

Higher-order corrections to the KdV equation may be derived in a straightforward manner. For instance, the third approximatetidirectional system (25), (26) with Hamiltonian & may be taken as the starting point. One subtracts impulse 11 given by (37) from the Hamiltonian, changes to the new variables r, s and restricts phase space to

A1 = {Ctl, u) E 2) :II tl - u IIH~cq,< ,u’} = {(r,s) E 2) :lI s llH~cso~~ &‘>.

The result is

In unscaled variables, the approximate equations of motion are

la 6fih

( >

1 a rt = --- 2ax Sr =-2E ($r* + fk + &rxxxx + )rj - [rrxlx),

ia s& St=-- 2ax 6s =O* ( >

384 W. Craig, M.D. Groves / Wave Motion I9 (1994) 36 7-389

The next higher-o?der approximation is obtained from the fourth approximate bidirectional system (27 ), (28) with Hamiltonian Hc. One carries out the usual steps, this time restricting phase space to

dz = {(?,a) E v :II r7 - u II 1f3(~)< P4) = {b-,s) E 27 4 s IIH~c~,,< $‘I.

The result is

fi - I^= Gj + 0(/P),

where

The approximate equations of motion are

la Sai rl = ---

2ax 6r ( > i a

= -335 (ir’ + $rxx + &rxxxx + fr: - [rrxlx + $rxxxxxx - frxrxxx

+ i [rrxxxlx + 3 [rrxlxxx - rr: + [r2rxlx),

ia SG o St=--

2ax 6s = * ( >

One can go on to higher orders with a little more work.

6. The KP equation

The KP equation is often considered a three-dimensional analogue of the KdV equation. It is derived from a small-amplitude, long-wave theory that uses the nonisotropic length scaling introduced in Section 4. One considers waves that are long in one horizontal direction and very long in the other. The KP equation is then obtained by the restriction to unidirectional wave motion. The objective of the present section is to focus this restriction on the Hamiltonian framework in a manner analogous to the derivation of the KdV equation described in Section 5.

The starting point is the second long-wave approximation in the nonisotropic Boussinesq scaling regime derived in Section 4 (Eqs. (30) and (3 1) ). This approximation is a Hamiltonian evolutionary system with the classical symplectic structure described in Section 2 and Hamiltonian fi = Es + O($), where fis is given by (29). The

first step is to make the change of variables (Q @ ) + (q, u) described in Section 5.1. The problem in the new variables is given by (33 ), (34) with

i?= J

[;gp’rl: + +:(rti + h) + ~p6DIx,‘b,,H2- @dx,] dx dz + O(P*).

SO

Here the usual stretched independent variables

x1 = PX, t1 = pt


(48)


have been introduced. This system is a Hamiltonian evolutionary system of the form ( 1) with structure map JB given by (35 1. There are two elementary Casimirs, namely

ij= J qdxdz, li = J

u dx dz.

SO SO

One must restrict phase space 2) to the subset

M={(q,u)~V:rj=ri=O)

in order to define the operator 0;’ appearing in the integrand of (48). In this restricted phase space JB may be inverted to define the symplectic 2-form

& = /(drlAD;i du) dx dz. J SO

One now shadows the steps in the derivation of the KdV equation. The first step is to nondimensionalise by setting g = h = 1. One then replaces fi with i) - 17, where impulse 17 is defined by

f& J

p4qu dx dz.

SO

This process corresponds to the introduction of new independent variables

A-2 = xi - t1, t2 = p2t,.

To focus upon the right-moving evolution one makes a further change of variables

r = i(tl+ u),

s= fw-u),

with corresponding scaled variables

rl = Tfir, si = +s,

in terms of which the system is given by (43), (44) with

-,dD,;‘h)Dx;‘(s~~) + &u6[Dx;1h)12} dx dz + O(,u*).

(49)

(50)

(51)

This system is a Hamiltonian evolutionary system defined on the restricted phase space

M = {(r,s) E 2, : i = i = 0).


The cosymplectic operator is JK~v, given by (47), and the symplectic 2-form is

&P= (drAD;‘dr- ds~D;‘d,r) dxdz. J so

The final step is to look for solutions where q is close to u in the H’ (So) norm. The Hamiltonians (48), (5 1) are defined in all of phase space M. Let A be an $-neighbourhood of q = u in the sense that

A = {(rl,u) E M :I1 tl- u II HI< ~‘1 = {(r,s) E M :I1 8 IL,qs,,~< ;P’>,

and restrict the phase space to A. All but the second, third and last terms in (51) are then O(,u*) and the Hamiltonian therefore reduces to

g-17 = Ekp +0(/P),

where

z&p = J { fbf - #%2 + fp6 Lox;’ (h 1 I’} dx dz.

SO

Returning to unscaled variables, one may write the system explicitly as

la B&p

( >

i a rt=-Zax 6r = -232 (;r2 + !rxx + 2D.F2(rzz)),

Here (52) is the KR equation

(52)

(53)

rt + irrx + irxxx + D;‘(rzr) = 0

in the variable r and (53) is an equation which says that in the present approximation there is no change in S. There are mathematical questions concerning the time of validity of the KP equation. These questions are

similar in nature to those posed for the KdV equation at the end of Section 5. At present there is no theorem to justify the KP approximation and the authors feel that work in this direction would be a worthwhile exercise.

Appendix. The Taylor expansion of the Dirichlet-Neumann operator

The goal of this section is to derive a recursion formula for the terms in the expansion of the Dirichlet-Neumann operator

G(V) E eGj(V) j=O

in n dimensions (n = 2,3). The fluid domain is the region

D,, = {by) :x E SO,Y E R, -h Q Y < r/(x)},

which is bounded below by the rigid flat bottom y = -h and above by the graph

S = { (X,Y) : x E So,Y = v(x)).

The elliptic boundary value problem to be solved is


Ag, = 0 in D,,

Q, = @ on s,

aY, -=0 my=-h, an

together with the appropriate periodic or asymptotic conditions on (p. There is a complete family of solutions to the problem: for k E UP-‘, with (k(’ = C k_f, define

pk(&y) = ei’x’k’cosh(lkI(y + h)). (.I)

These functions have the boundary values @k = eicx’k) cosh( Jkj (q(x) + h) ), and are compatible with the asymptotic conditions as IX I + co, or with conditions of periodicity for appropriately chosen wavevectors k. (If periodic boundary conditions are stated so that Q, (X + y, y ) = Q (x, y ) for all y in a lattice r, then one chooses k E r’, where r’ is the dual lattice to r.)

Because

G(q)@’ = R 31 an Jk)l’

the fUnCtiOnS pk Satisfy the equation

@k %’ agk ay - -. - 1 ax ax &

t.21

Substituting for pk from (. 1) and expanding the hyperbolic functions and Dirichlet-Neumann operator in (.2), one obtains the identity

c Wh)’ j! sinh(lklh)

jevm

- is e kcosh([klh)) eicx”)

+ c v (cosh(lklh) - ig . k sinh(lklh)) eicx”) j&l ’

= ($GttU)) [~~cosh(lkjh)ei’x’k’ +E@!$sinh([klh)ei(X’k’).

The recursion formula is obtained by identifying terms of the same degree in rl. For j = 0 the result is

Gaei’x’k) = Ikl tanh(hlkI) eicx’k) .

Using Fourier analysis one finds that for a general (sufficiently well-behaved) function @, GsQi is given by

G&(x) = IDI tanh(hlDI)@ (r). (.3)

Higher or*r terms in the expansion are derived in a similar manner. For a positive, even j one has

Gj(q) = $ (#lDI’+‘tanh(hlDI) -i~(4’).DIDI’-‘tanh(hlDI))

- (.4)

while for odd j one has

388 W. Craig, M.D. Groves / Wave Motion I9 (1994) 367-389

-c 1 -GL (rj)t$-‘IDI’-f.

L<j,L odd (j - e )l (5)

Equations (.3)- (.5 ) constitute the recursion formula for the terms Gj (q ). A similar technique has been used to obtain an expansion of the Dirichlet-Neumann operator for the harder

problem of water waves in a uniform channel of arbitrary cross-section [ 341.

Acknowledgements

One author (MDG) wishes to record his thanks for an SERC Research Studzentship, and the other (WC) gratefully acknowledges the support of NSF Grant DMS-8920624 and an Alfred P. Sloan Research Fellowship. The paper was written during an extended visit by WC to the Mathematical Institute, Oxford University.

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Hamiltonian long-wave approximations to the water-wave problem

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