European Journal of Mechanics B/Fluids 32 (2012) 22–31

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European Journal of Mechanics B/Fluids

journal homepage: www.elsevier.com/locate/ejmflu

Hamiltonian higher-order nonlinear Schrödinger equations for broader-bandedwaves on deep water

Walter Craig a, Philippe Guyenne b,∗, Catherine Sulem c

a Department of Mathematics, McMaster University, Hamilton, ON L8S 4K1, Canadab Department of Mathematical Sciences, University of Delaware, Newark, DE 19716-2553, USAc Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada

a r t i c l e i n f o

Article history:Received 10 January 2011Received in revised form4 June 2011Accepted 22 September 2011Available online 1 October 2011

Keywords:Water wavesNonlinear Schrödinger equationHamiltonian systemsModulation theoryDirichlet–Neumann operatorSymplectic integrators

a b s t r a c t

Starting from the Hamiltonian formulation of the water wave problem and using the approach recentlydeveloped by Craig et al. (2010) [8], we derive Hamiltonian versions of the higher-order nonlinearSchrödinger equations for broader-banded deep-water waves, originally proposed by Trulsen and co-workers (1996, 2000) [4,5]. A Benjamin–Feir stability analysis is performed and shown to be in goodagreement with previous work. Numerical simulations using a symplectic time integration scheme arealso presented to illustrate the performance of these new models with regards to their conservative andBenjamin–Feir stability properties.

© 2011 Elsevier Masson SAS. All rights reserved.

1. Introduction

The nonlinear Schrödinger (NLS) equation is a canonical modelfor describing theweakly nonlinearmodulation of a train of surfacegravity waves. It is accurate up to O(ε3) and is valid for waves ofbandwidth O(ε), where ε is a small parametermeasuring thewavesteepness. Besides inherent limitations related to the assumptionof small wave steepness, the NLS equation also exhibits anunbounded region of Benjamin–Feir instability, in the case of two-dimensional sideband perturbations, which extends outside theregime of a narrow-banded spectrum. As a result, energy initiallycontained at low wavenumbers can leak to higher ones, as shownin numerical simulations of Martin and Yuen [1].

These limitations have prompted a number of initiatives inorder to extend the range of applicability of the NLS equation. Fordeep-water waves, Dysthe [2] considered terms of up to O(ε4)(see [3] for the finite-depth case). His analysis reveals contributionsfrom themean flow induced by radiation stresses of themodulatedwavetrain. This mean flow causes a local Doppler shift in themain direction of wave propagation, which results in improved

∗ Corresponding author. Tel.: +1 302 831 8664; fax: +1 302 831 4511.E-mail address: guyenne@math.udel.edu (P. Guyenne).

0997-7546/$ – see front matter© 2011 Elsevier Masson SAS. All rights reserved.doi:10.1016/j.euromechflu.2011.09.008

stability properties. Subsequently,motivated by the fact that oceanwave spectra are usually not as narrow banded as assumed bythe NLS equation, Trulsen and Dysthe [4] derived a high-ordermodel similar to Dysthe’s, which allows for waves of slightlylarger bandwidth, O(ε1/2). Their model essentially retains thesame accuracy in nonlinearity as Dysthe’s equation, but exhibitsadditional higher-order linear dispersive terms. Trulsen et al. [5]then took this idea further by combining the exact linear dispersionrelation for deep-water waves with the cubic nonlinear terms ofDysthe’s equation. This approach allows the exact linear dispersiveterm to be efficiently computed by a pseudo-spectral method,while retaining the relative simplicity of Dysthe’s equation. Asignificant improvement on stability properties was observed incomparison with McLean’s results on exact Stokes waves [6].More specifically, Trulsen et al.’s analysis [5] reveals a boundedBenjamin–Feir instability region which prevents energy fromleaking to high wavenumbers in their model.

While these earlier higher-order versions of the NLS equationhave been applied with reasonable success to modeling avariety of wave phenomena, including four-wave interactionsin applications to ocean wave spectra and rogue waves, theyshare a fundamental shortcoming: they are not Hamiltonianpartial differential equations, despite the fact that they representapproximations to the Euler equations which can be written as aHamiltonian system [7]. This in part motivated the recent work of

W. Craig et al. / European Journal of Mechanics B/Fluids 32 (2012) 22–31 23

Craig et al. [8,9] who proposed a systematic Hamiltonian approachto nonlinear wavemodulation. In particular, these authors derivedHamiltonian versions of Dysthe’s equation for gravity water waveson both finite and infinite depth. The present paper takes thisidea further by proposing Hamiltonian counterparts to the modelsderived by Trulsen and coworkers [4,5], using the method recentlydeveloped by Craig et al. [8]. These new Hamiltonian models, notonly possess a high degree of accuracy, but are also consistentwith theHamiltonian formulation of the full waterwave problem1.In addition to presenting their derivation, we also analyze theirproperties with regards to the Benjamin–Feir stability of a uniformwavetrain (i.e. a Stokes wave). These stability results are testedagainst numerical simulations using a fourth-order symplecticscheme for time integration. To our knowledge, this is the first timethat results are reported on applications of this type of symplecticintegrators to Hamiltonian higher-order NLS equations for waterwaves.

The remainder of the paper is organized as follows. InSection 2,wepresent themathematical formulation of the problemincluding the Hamiltonian formulation of the equations of motion.Sections 3 and 4 describe the main steps in our Hamiltonianperturbation method, and Sections 5 and 6 give the derivation ofour Hamiltonian models. The Benjamin–Feir stability analysis ofthese models is presented in Section 7, and numerical results arethen discussed in Section 8. Finally, concluding remarks are givenin Section 9.

2. Hamiltonian formulation and Dirichlet–Neumann operator

We consider the evolution of a free surface y = η(x, t) on topof an infinitely deep fluid

S(η) = (x, y) ∈ Rn−1× R : −∞ < y < η(x, t),

under the influence of gravity. Here, (x, y) denote the horizontaland vertical coordinates respectively, t is time, and n = 2 or 3 isthe space dimension. The fluid is assumed to be incompressible,inviscid and the flow is irrotational, so that the free-surfaceelevation η(x, t) and the velocity potential ϕ(x, y, t) satisfy theboundary value problem

∇2ϕ = 0 in S(η), (1)

∂tη + ∂xη · ∂xϕ − ∂yϕ = 0 at y = η(x, t), (2)

∂tϕ +12|∇ϕ|

2+ gη = 0 at y = η(x, t), (3)

∂yϕ → 0 as y → −∞, (4)

where g denotes the acceleration due to gravity and∇ = (∂x, ∂y)⊤.

Following Craig and Sulem [12], we can reduce the dimen-sionality of the classical formulation (1)–(4) for the water waveproblem by considering surface quantities as unknowns. This canbe accomplished by introducing the Dirichlet–Neumann operator(DNO)

G(η)ξ = (−∂xη, 1)⊤ · ∇ϕ|y=η, (5)

which takes Dirichlet data ξ(x, t) = ϕ(x, η(x, t), t) at the freesurface, solves the Laplace equation (1) for ϕ with boundarycondition (4), and returns the corresponding Neumann data(i.e. the normal fluid velocity at the free surface).

1 After we submitted this paper, we learnt about the recent work of Gramstadand Trulsen [10] who derived a Hamiltonian form of the modified NLS equationfor gravity waves on arbitrary depth, starting from Krasitskii’s version [11] ofZakharov’s equation.

In terms of ξ and G(η)ξ , Eqs. (1)–(4) reduce to

∂tη = G(η)ξ, (6)

∂tξ = −gη −1

2(1 + |∂xη|2)

× [|∂xξ |2− (G(η)ξ)2 − 2(∂xξ · ∂xη)G(η)ξ

+ |∂xξ |2|∂xη|

2− (∂xξ · ∂xη)2], (7)

which are Hamiltonian equations in Zakharov’s formulation ofthe water wave problem [7,12,13]. These can be expressed in thecanonical form

∂t

ηξ

=

0 1

−1 0

δηHδξH

, (8)

for the conjugate variables η and ξ , with the Hamiltonian

H =12

[ξG(η)ξ + gη2

]dx. (9)

Eq. (9) can be thought of as the total energy of the system, withthe first and second terms representing the kinetic and potentialenergies respectively.

It has been shown that the DNO is an analytic function of ηprovided the free surface is sufficiently regular [14], which impliesthat the DNO can be written in terms of a convergent Taylor seriesexpansion

G(η) =

∞j=0

Gj(η), (10)

where the Taylor polynomials Gj can be determined recur-sively [12]. However, only contributions of up to second order inη, i.e.

G0 = |Dx|,

G1 = Dxη · Dx − G0ηG0,

G2 = −12(|Dx|

2η2G0 + G0η2|Dx|

2− 2G0ηG0ηG0),

are needed for the purposes of the present study, as they includeall the contributions relevant to four-wave interactions [15].Note that Dx = −i∂x (so its Fourier symbol is k) and, in thecase of a (constant) finite depth h, the only modification to thisformulation is G0 = |Dx| tanh(h|Dx|) [12,13]. The reader may referto [16–21,13] for applications of this formulation to long-waveperturbation calculations as well as direct numerical simulationsof nonlinear waves on both uniform and variable depth.

3. Canonical transformations and modulational Ansatz

Following Craig et al. [22,8,9,23], our Hamiltonian approach forderiving envelope models involves canonical transformations thatapproximate the original Hamiltonian of the system and changethe corresponding symplectic structure. First, we introduce thenormal modes (z, z,η,ξ) defined by

η =1

√2a−1(Dx)(z + z) +η, η = P0η, (11)

ξ =1

√2i

a(Dx)(z − z) +ξ, ξ = P0ξ, (12)

where

a(Dx) = 4

gG0

,

and (η,ξ) are the zeroth modes representing the mean flow. Thesymbol . stands for complex conjugation, and P0 is the projection

24 W. Craig et al. / European Journal of Mechanics B/Fluids 32 (2012) 22–31

that associates to (η, ξ) their zeroth-frequency components. Wesplit the zeroth modes from the higher ones in this decompositionbecause otherwise a−1(0) = 0 and so the transformation (η, ξ) →

(z, z) is singular for k = 0. As a result, system (8) becomes

∂t

zzηξ =

0 −i(I − P0) 0 0i(I − P0) 0 0 0

0 0 0 P00 0 −P0 0

δzH

δzHδηHδξH

, (13)

where I is the identity operator.The next step is to introduce the modulational Ansatz

z = εu(X, t)eik0·x, z = εu(X, t)e−ik0·x, (14)η = εα+1/2η1(X, t), ξ = εαξ1(X, t), (15)

which is to say that we look for solutions in the form ofmonochromatic waves with carrier wavenumber k0 ∈ Rn−1

+ \ 0and with slowly varying complex envelope u depending on X =

ε1/2x. As suggested by Trulsen and Dysthe [4], this choice of longspatial scale allows for waves of slightly larger bandwidth thanassumed by the NLS and Dysthe equations. The exponent α ≥

1 is to be determined by the subsequent asymptotic procedure,and ε = O(|k0|a0) ≪ 1 is a small parameter measuring thewave steepness (a0 is a typical wave amplitude). In (15), there isa difference by an exponent 1/2 in the power of ε between themean fieldsη andξ because we anticipate thatη ∼ ∂xξ similarlyto the long-wave regime [17,22,19]. The corresponding equationsof motion are given by

∂t

uuη1ξ1

=

0 −iε(n−5)/2I′ 0 0

iε(n−5)/2I′ 0 0 00 0 0 ε(n−2−4α)/2

0 0 −ε(n−2−4α)/2 0

×

δuHδuHδη1Hδξ1H

, (16)

where I′ is the identity on the class of functions u(X), and thefinal 2 × 2 block retains essentially the standard symplectic formon the two-dimensional space of functions (η1,ξ1).

Note the successive changes in the symplectic structure of thesystem, as represented by the different coefficient matrices onthe right-hand side of (8), (13) and (16). Further details on thesecanonical transformations can be found in [22,8].

4. Expansion and reduction of the Hamiltonian

The expression of the Hamiltonian (9) is also transformedthrough the changes of variables (11)–(12) and (14)–(15). The firsttransformation diagonalizes the quadratic (i.e. linear) part of theHamiltonian, so as to exhibit more clearly the natural frequenciesof the system. The second one introduces the small parameter εand, together with the Taylor series expansion of the DNO, thisallows us to expand H in powers of ε. For example, the Fouriermultiplier Dx = ε1/2DX when acting on functions of X only,while Dx = ±k0 + ε1/2DX when acting on functions of the formu(X)e±ik0·x [23].

Moreover, since we look for solutions in the form of multiplescale functions, with the slowly varying components being thefocus of our attention, further simplifications can be achieved

by only retaining resonant terms in the Hamiltonian. Thishomogenization (or averaging) procedure is based on the scaleseparation result of Craig et al. [19], which implies that terms withfast oscillations essentially homogenize to zero and thus do notcontribute to the effective Hamiltonian. More specifically, if g(x)is a periodic function and f (X) is a Schwartz class function, then

g

X√

ε

f (X)dX = E(g)

f (X)dX + O(εN),

for any N > 0, with E(g) being the average value of g overa fundamental domain. A more precise statement of this resulttogether with its proof can be found in [19]. In the present setting,the homogenized coefficients are of the form

E(g) = E(eik0n·x) =

1 if n = 0,0 if n = 0,

which may be interpreted as resonance conditions for n-waveinteractions.

Starting from the decompositionH = H2 + H3 + H4 + · · · ,

where

H2 =12

(ξG0ξ + gη2)dx, H3 =

12

ξG1ξdx,

H4 =12

ξG2ξdx,

we obtain, after transformations and simplifications,

H2 = ε(5−n)/2

uω(k0 + ε1/2DX )u +12ε2α−3/2ξ1|DX |ξ1

+g2ε2α−1η2

1

dX + · · · , (17)

H3 =i2ε(6+2α−n)/2

[2k0|u|2 + ε1/2(uDXu + uDXu)] · DXξ1

+12ε1/2 k0

|k0|· (uDXu − uDXu)|DX |ξ1 dX + · · · , (18)

H4 =14ε(9−n)/2

|k0|3

|u|4 +32ε1/2 k0

|k0|2· |u|2(uDXu

+ uDXu)dX + · · · , (19)

where ω(Dx) = (gG0)1/2

= (g|Dx|)1/2 is the Fourier multiplier

representing the exact linear dispersion relation of the problem.

5. O(ε5/2) model

If we now expandω(k0+ε1/2DX ) up toO(ε5/2) so as to considerthe same order of approximation as in [4], and collect the variousterms, we find

H = ε(5−n)/2

u2

ω(k0) + ε1/2∂kω(k0) · DX

+ε

2∂2kjklω(k0)D2

XjXl + · · · +ε5/2

120∂5ω(k0)D5

u

+ c.c. +12ε2α−3/2ξ1|DX |ξ1 +

g2ε2α−1η2

1

+i2εα+1/2

[2k0|u|2 + ε1/2(uDXu + uDXu)] · DXξ1+

i4εα+1 k0

|k0|· (uDXu − uDXu)|DX |ξ1

+ε2

4|k0|3

|u|4 +

32ε1/2 k0

|k0|2· |u|2(uDXu + uDXu)

dX

+ . . . , (20)

W. Craig et al. / European Journal of Mechanics B/Fluids 32 (2012) 22–31 25

where ‘c.c.’ stands for the complex conjugate of all the precedingterms on the right-hand side of the equation. Dominant balancesuggests that 2α − 3/2 = 5/2 and thus α = 2, which leads to

ε(n−5)/2H =

u2

ω(k0) + ε1/2∂kω(k0) · DX

+ε

2∂2kjklω(k0)D2

XjXl + · · · +ε5/2

120∂5ω(k0)D5

u

+ c.c. +ε2

4|k0|3|u|4 + ε5/2

12ξ1|DX |ξ1

+ i|u|2k0 · DXξ1 +38|k0|k0 · |u|2(uDXu

+ uDXu)dX + O(ε3). (21)

In this equation, the Einstein summation convention is used forrepeated indices (j, l = 1, . . . , n − 1), and the fifth-orderderivative term is expressed in the short multi-index notation. Theso-obtained Hamiltonian can be further reduced by subtractingmultiples of the conserved wave action

M = ε(5−n)/2

|u|2dX, (22)

and of the conserved impulse (or momentum)

I =

η∂xξ dx,

= ε(5−n)/2

k0|u|2 +ε1/2

2(uDXu + uDXu)

+ iε3η1DXξ1 dX, (23)

so that the reduced form is

ε(n−5)/2H= ε(n−5)/2

H − ∂kω(k0) · I − [ω(k0) − k0 · ∂kω(k0)]M,

=ε

2

u2

∂2kjklω(k0)D2

XjXl + · · · +ε3/2

60∂5ω(k0)D5

u

+ c.c. +ε

2|k0|3|u|4 + ε3/2

ξ1|DX |ξ1 + 2i|u|2k0 · DXξ1+

34|k0|k0 · |u|2(uDXu + uDXu)

dX + O(ε3). (24)

IntroducingM is equivalent to saying that our approximation of theproblem is phase invariant (as is typically the case for the NLS andDysthe equations). Subtracting I from H is equivalent to changingthe coordinate system into a reference frame moving with thegroup velocity ∂kω(k0) [22,8].

From (16), the corresponding equations of motion read

2i∂τu = ∂2kjklω(k0)D2

XjXlu + · · · +ε3/2

60∂5ω(k0)D5u

+ ε|k0|3|u|2u + ε3/2(2ik0 · uDXξ1 + 3|k0|k0· |u|2DXu) + O(ε2), (25)

ε∂τ η1 = |DX |ξ1 − ik0 · DX |u|2 + O(ε1/2), (26)

ε∂τ ξ1 = O(ε1/2), (27)

where τ = εt is a slow time. Noting that the mean fieldξ1 onlyappears at order O(ε3/2) in (25), we can solve (26) forξ1 at leadingorder,ξ1 = i|DX |

−1k0 · DX |u|2 + O(ε1/2), (28)

and plug this expression in (25). The resulting closed equation forthe complex envelope u,

2i∂τu = −∂2kjklω(k0)∂2

XjXlu + · · · − iε3/2

60∂5ω(k0)∂5u

+ ε|k0|3|u|2u + ε3/2(2uk0jk0 l|DX |−1∂2

XjXl |u|2

− 3i|k0|k0 · |u|2∂Xu), (29)

is a Hamiltonian version of the model derived by Trulsen andDysthe [4]. It can be cast into the symplectic form

∂τu = −iδuH, (30)

with the Hamiltonian

H =12

u2

−∂2

kjklω(k0)∂2XjXl + · · · − i

ε3/2

60∂5ω(k0)∂5

u

+ c.c. +ε

2|k0|3|u|4 + ε3/2

32|k0|k0 · |u|2ℑ(u∂Xu)

− k0jk0 l(∂Xj |u|2)|DX |

−1∂Xl |u|2

dX, (31)

whereℑ denotes the imaginary part. This closed-formHamiltonianresults from (24) by substitutingξ1 with (28).

We remark that, in the finite-depth case, dominant balancewould likely imply a smaller value of α, which means that themean flowwould occur at a lower order in the asymptotic analysis.Derivations in this case for waves of bandwidth O(ε) can be foundin [8].

6. O(ε5/2) model with exact linear dispersion

As suggested by Trulsen et al. [5], a better approximation canbe achieved by expanding the nonlinear terms in (9) up to O(ε5/2)as previously, while keeping the linear dispersion relation exact. Inthe present framework, the resulting envelope equation is

2i∂τu =2ε[ω(k0 + ε1/2DX ) − ω(k0)]u + ε|k0|3|u|2u

+ ε3/2(2uk0jk0 l|DX |−1∂2

XjXl |u|2

− 3i|k0|k0 · |u|2∂Xu), (32)

which can be viewed as aHamiltonian version of themodel derivedin [5], and the corresponding Hamiltonian is given by

H =12

2εu[ω(k0 + ε1/2DX ) − ω(k0)]u +

ε

2|k0|3|u|4

+ ε3/232|k0|k0 · |u|2ℑ(u∂Xu)

− k0jk0 l(∂Xj |u|2)|DX |

−1∂Xl |u|2

dX . (33)

Note that the extra term proportional to ω(k0) in (33) is due to thereductionH → H−ω(k0)M , and the symplectic structure (30) stillholds true for this case.

Eq. (32) looks very similar to that derived by Trulsen et al. [5] inits general form, with the exception that a higher-order nonlinearterm like k0 · u2∂Xu is absent, and here we are also able toexplicitly close the equation by solving for the mean flow. Thesame observations apply to (29) in comparison with the model ofTrulsen and Dysthe [4]. It is possible to recover the original modelsfrom their Hamiltonian counterparts by a change of variables,as shown in [8] for the Hamiltonian Dysthe equation, but thistransformation is not canonical. Eq. (32) also closely resembles theHamiltonianmodel of Gramstad and Trulsen [10] in its deep-waterlimit, although these authors did not express their equation in anexplicitly closed form.

26 W. Craig et al. / European Journal of Mechanics B/Fluids 32 (2012) 22–31

a b

c d

Fig. 1. Comparison of instability regions for A0 = 0.1 and ε = 1 between (a) the NLS equation, (b) the Hamiltonian Dysthe equation, (c) model (29) and (d) model (32).

7. Stability of Stokes waves

Both models (29) and (32) admit uniform wavetrain solutions

u0(τ ) = A0e−i2 εk30A

20τ ,

where A0 is a real constant, which correspond to progressiveStokes waves. In this section, we analyze the modulational orBenjamin–Feir stability of these solutions, i.e. their linear stabilitywith respect to sideband perturbations. For this purpose, weconsider the general three-dimensional case n = 3 such thatx = (x1, x2)⊤ ∈ R2, and we assume that k0 points in the x1-direction.

Examining model (29) first, if we insert a perturbed solution ofthe form

u(X, τ ) = u0(τ )[1 + B(X, τ )],

where

B(X, τ ) = B1eΩτ+i(λX1+µX2) + B2eΩτ−i(λX1+µX2),

and B1, B2 are complex coefficients, we find that the conditionℜ(Ω) = 0 for instability yields

εA20

gk0

λ2

2− µ2

+1596

ελ4

k20−

158

ελ2µ2

k20+

924

εµ4

k20

×

k0 − 2ε1/2 λ2

λ2 + µ2

−g4k30

λ2

2− µ2

+1596

ελ4

k20−

158

ελ2µ2

k20+

924

εµ4

k20

2

> 0, (34)

where ℜ denotes the real part. This somewhat straightforwardbut tedious calculation is similar to those presented in [3,2,24,4,5],therefore we skip the details and only show the final result for thereader’s convenience.

Applying the same strategy tomodel (32),we find that sidebandinstability occurs when

2ε2k20A20[D(λ, µ) + D(−λ, µ)]

k0 − 2ε1/2 λ2

λ2 + µ2

− [D(λ, µ) + D(−λ, µ)]2 > 0, (35)

which is expressed in terms of the exact linear multiplier

D(λ, µ) =gk0 −

g[(k0 + ε1/2λ)2 + εµ2]1/2.

As expected, both (34) and (35) indicate that the linear dispersiveterms as well as the mean-flow term play an important role on thegrowth of sideband perturbations. In particular, we clearly noticethe ‘Doppler shift’ relative to the carrierwavenumber k0, due to themean flow.

Fig. 1 shows the instability regions enclosed by the zero-levelcontour of conditions (34) and (35). As a reference, we also in-clude the plots of the instability regions for the cubic NLS andHamiltonian Dysthe equations as derived in [8]. To allow fordirect comparison with existing results [6,4,5], we choose theparameter values A0 = 0.1, g = 1, k0 = 1 and ε = 1.The latter choice of value for ε allows for a more suitable scaling-independent inter-comparison between the different asymp-totic models under consideration. Overall, we observe strongsimilarities with previously published results based on otherHamiltonian [10] and non-Hamiltonian models [5]. A clear im-provement from (29) and (32) over the NLS and Dysthe equa-tions is that the neutral stability curves are no longer straightlines. In particular, the instability region for (32) is localized nearthe origin and takes an arched shape connecting back to theλ-axis, which closely resembles McLean’s class I results on ex-act Stokes waves of small amplitude [6]. This supports the ideathat linear dispersion strongly affects the Benjamin–Feir instabilityprocess.

W. Craig et al. / European Journal of Mechanics B/Fluids 32 (2012) 22–31 27

8. Numerical results

Although the analysis and derivations presented in theprevious sections apply to the general three-dimensional case,for convenience here, we will only show two-dimensional (n =

2) numerical simulations. In this situation, models (29) and (32)simplify respectively into

2i∂τu = −∂2k ω(k0)∂2

Xu + iε1/2

3∂3k ω(k0)∂3

Xu

+ε

12∂4k ω(k0)∂4

Xu − iε3/2

60∂5k ω(k0)∂5

Xu

+ εk30|u|2u − ε3/2k20(2u|DX | |u|2 + 3i|u|2∂Xu), (36)

whose Hamiltonian is

H =12

u2

−∂2

k ω(k0)∂2X + i

ε1/2

3∂3k ω(k0)∂3

X

+ε

12∂4k ω(k0)∂4

X − iε3/2

60∂5k ω(k0)∂5

X

u

+ c.c. +ε

2k30|u|

4+ ε3/2k20

32|u|2ℑ(u∂Xu)

− |u|2|DX ||u|2

dX, (37)

and

2i∂τu =2ε[ω(k0 + ε1/2DX ) − ω(k0)]u + εk30|u|

2u

− ε3/2k20(2u|DX | |u|2 + 3i|u|2∂Xu), (38)

with the Hamiltonian

H =12

2εu[ω(k0 + ε1/2DX ) − ω(k0)]u +

ε

2k30|u|

4

+ ε3/2k20

32|u|2ℑ(u∂Xu) − |u|2|DX | |u|2

dX . (39)

Note that now X = X1 ∈ R and k0 > 0.The purpose of this section is twofold. First, we want to

numerically check that models (29) and (32) derived in Sections 5and 6 conserve their respective Hamiltonians in time. Second,we want to numerically test and validate the stability analysispresented in Section 7. Since this analysis only concerns linearstability while the problem is nonlinear, it is of interest toexamine e.g. the validity of these results for long time intervals.Furthermore, we take this opportunity to introduce a symplecticnumerical scheme for time integration of the two proposedmodels, motivated by the fact that they are Hamiltonian. Weemphasize however that our intention is not to claim that thisscheme is crucial for correctly simulating these Hamiltonianequations, nor is it superior to other (non-symplectic) schemes.This debate is beyond the scope of the present paper. Rather, wewant to offer a possible choice of symplectic time integrator whichis both numerically efficient and accurate. Details are given below.

8.1. Numerical methods

For space discretization, we use a pseudospectral methodassuming periodic boundary conditions in X [12,21,25]. Morespecifically, the complex envelope u is approximated by atruncated Fourier series. Spatial derivatives and nonlocal Fouriermultipliers are evaluated in Fourier space, while nonlinearproducts are calculated in physical space, on a grid of N equallyspaced points. For example, the term ω(k0 + ε1/2DX )u in (38) canbe efficiently computed by

F −1[

g|k0 + ε1/2λ|F (u)],

using the fast Fourier transform F . Aliasing errors are removed byzero-padding in Fourier space, meaning that for the calculation ofthe nonlinear terms, the size of the solution’s spectrum is extendedby a factor of 2 and the extra modes are set to zero.

Time integration of (36) and (38) is performed in Fourier space,so that the linear terms can be solved exactly by the integratingfactor technique. For illustration, let us consider model (38) in itsFourier form

∂τu = Lu + N (u),where

L = −iε

g|k0 + ε1/2λ| −

gk0,

is the Fourier multiplier of the linear part (L = −L), N (u) is thenonlinear part andu(λ, τ ) = F (u(X, τ )). Thenmaking the changeof variablesu(λ, τ ) = eLτv(λ, τ ), (40)

leads to the following nonlinear evolution equation forv,∂τv = e−Lτ N (eLτv) = N (v). (41)

Because Eq. (40) is a canonical transformation [22,8,13], throughwhich

∂τ

uu

=

0 −iI′

iI′ 0

δuHδuH

,

is transformed into

∂τ

vv

=

e−Lτ 00 eLτ

0 −iI′

iI′ 0

e−Lτ 00 eLτ

⊤ δvHδvH

,

=

0 −iI′

iI′ 0

δvHδvH

,

therefore Eq. (41) for v is also Hamiltonian with the samesymplectic structure (30) as for u andu. That (40) is a canonicaltransformation should be expected since all the linear terms in (38)have counterparts in the correspondingHamiltonian (39), and thus(40) can be thought of as a phase change (in Fourier space) whichfurther renormalizes the Hamiltonian by subtracting off the linearcontributions.

We integrate (41) in time using a symplectic fourth-order(2-stage) Gauss–Legendre Runge–Kutta scheme [26],vn+1

=vn + 1τ [b1 N (v(1)) + b2 N (v(2))],v(1)=vn + 1τ [a11 N (v(1)) + a12 N (v(2))],v(2)=vn + 1τ [a21 N (v(1)) + a22 N (v(2))], (42)

for the solutionvn+1 at τn+1 = τn + ∆τ , where 1τ is the constanttime step and

a11 = a22 =14, a12 =

14

+

√36

, a21 =14

−

√36

,

b1 = b2 =12, c1 =

12

+

√36

, c2 =12

−

√36

.

By inverting (40), we can rewrite (42) in terms ofu asun+1= eL1τun

+ 1τeL1τ[b1e−c1L1τ N (ec1L1τu(1))

+ b2e−c2L1τ N (ec2L1τu(2))], (43)u(1)= un

+ 1τa11e−c1L1τ N (ec1L1τu(1))

+ 1τa12e−c2L1τ N (ec2L1τu(2)), (44)u(2)= un

+ 1τa21e−c1L1τ N (ec1L1τu(1))

+ 1τa22e−c2L1τ N (ec2L1τu(2)). (45)

At each time step, the values of u(1) and u(2) required in (43)to update un+1 are obtained by solving the nonlinear system

28 W. Craig et al. / European Journal of Mechanics B/Fluids 32 (2012) 22–31

Fig. 2. Comparison of instability regions for A0 = 0.15, ε = 1 and µ = 0 (n = 2)between the NLS equation (dotted-dashed line), the Hamiltonian Dysthe equation(dashed line), model (36) (dotted line) and model (38) (solid line). The instabilitycurves for the latter two models are given by conditions (34) and (35).

(44)–(45). This is accomplished through fixed point iteration withthe initial guess foru(1) andu(2) given by the solutionun at time τn.A similar scheme was used in [13] to solve the full Eqs. (6)–(7). Forall the applications shown in the present paper, three iterationswere typically needed to solve the nonlinear system given aconvergence tolerance of 10−8 on the relative error. Thiswas foundto be a good compromise between accuracy and computationalcost.

We point out that the numerical methods for space discretiza-tion and time integration, as described above, can be readily ex-tended to the three-dimensional case (n = 3).

8.2. Discussion of results

For our two-dimensional simulations, we non-dimensionalizethe equations according to Stokes wave theory in deep water bymultiplying lengths by k0 and multiplying times by ω(k0) so that

both g = 1 and k0 = 1. Having the test on Benjamin–Feirinstability in mind, we start our computations with the perturbedsolution

u(X, 0) = A0[1 + ap cos(λpX)].

Since the main goal of this section is to illustrate properties ofmodels (36) and (38) alongwith the performance of the symplectictime integrator,wewill restrict our attention to the caseA0 = 0.15,ε = 1, ap = 0.01 and λp = 0.2. Because of our choice of non-dimensionalization, specifying A0 ≪ 1 while fixing ε = 1 isequivalent to specifying ε ≪ 1 while fixing A0 = 1. The initialwave steepness is measured by the parameter A0.

We have typically observed that smaller values of A0 and apinduce slower evolutions in time (and thus longer computations),while larger values induce faster evolutions, quickly leadingto higher-frequency physical/numerical instabilities beyond theBenjamin–Feir regime due to the higher nonlinearities involved.Therefore, the choice of A0 = 0.15 (and ap = 0.01)was found to bea good compromise for the purpose of our numerical illustrations.The value λp = 0.2 corresponds to the most unstable disturbanceas shown in Fig. 2. For comparison again,we also plot the instabilitycurves for theNLS andHamiltonianDysthe equations. Note that thecurves for (36) and (38) are indistinguishable at the graphical scaleof Fig. 2.

Figs. 3 and 4 show the time evolution of the relative errors onM and H up to τ = 2500, whereM0 and H0 are the initial values atτ = 0. We used a computational domain of length L = 20π withspatial resolution N = 256 and time step 1τ = 10−3. The valueof 1τ is selected such that it is much smaller than the smallestlinear period allowed by the chosen spatial resolution. Overall,bothM andH are verywell conserved by the twomodels, althoughthe corresponding errors exhibit a tendency to grow in time. Thisgrowth is likely due to the accumulation of numerical errors, whichis aggravated by the development of the Benjamin–Feir instability,and is more pronounced for model (38), especially with regards tothe conservation of H .

Fig. 5 depicts the time evolution of the normalized amplitudesfor the fundamental and sideband harmonics, |u(0)| and |u(±λp)|.

a b

Fig. 3. Relative error on wave actionM as a function of time for A0 = 0.15, ε = 1, ap = 0.01 and λp = 0.2. Left panel: model (36). Right panel: model (38).

a b

Fig. 4. Relative error on Hamiltonian H as a function of time for A0 = 0.15, ε = 1, ap = 0.01 and λp = 0.2. Left panel: model (36). Right panel: model (38).

W. Craig et al. / European Journal of Mechanics B/Fluids 32 (2012) 22–31 29

Fig. 5. Normalized harmonics as a function of time for A0 = 0.15, ε = 1,ap = 0.01 and λp = 0.2. Fundamental |u(0)| (thick solid line). Lower sideband|u(−λp)| (dashed line). Upper sideband |u(λp)| (thin solid line). Top panel: model(36). Bottom panel: model (38).

Formodel (36), the near-recurring exchange of energy between thefundamental and sidebands is consistent with previous observa-tions of the Benjamin–Feir instability for waves of relatively smallamplitude and bandwidth (e.g. [24,27,1,13]). The asymmetric evo-lution of the two sidebands, with the lower sideband being moreexcited than the upper one, is also a well-known feature of thephenomenon. It is mainly due to the higher-order nonlinear term|u|2∂Xu in the equations, which has a skewness effect on both thephysical and spectral aspects of the solution. In contrast, the resultsfor model (38) show a more irregular pattern, which may be ex-plained by the fact that the exact linear dispersion combined withthemoderately small wave steepness allows the Benjamin–Feir in-stability to cause sufficient wave modulations to trigger higher-wavenumber instabilities. This superposition of instabilities couldthen result in the observed irregular behavior.

Finally, Figs. 6 and 7 show snapshots of the envelopemagnitude |u| together with the corresponding (rescaled) free-surface elevation ε−1η. The latter can be determined from theenvelope u using the transformation (11) and, in the presentnumerical setting, it can be easily computed as

η(X, τ ) =ε

√2

F −1

4

|k0 + ε1/2λ|

gu eik0X/

√ε+ c.c.

. (46)

This expression neglects the mean field η which does notcontribute at the order of approximation considered here, but itexactly accounts for all the contributions from the higher sidebandharmonics since it includes the exact expression of G0. Overall, forboth models (36) and (38), the solution develops strong amplitudemodulations as a result of the Benjamin–Feir instability.We clearlysee the development of the left–right asymmetry in the profile of

Fig. 6. Snapshots of the envelope magnitude |u| (thick solid line) and free-surfaceelevation ε−1η (thin solid line) for model (36) at (a) τ = 0, (b) 799.967, (c) 949.961,(d) 1366.611 and (e) 1866.591, for A0 = 0.15, ε = 1, ap = 0.01 and λp = 0.2.

|u|, especially at the initial stages of the wave evolution, whichis related to the skewness effect as discussed earlier. The moreunstable behavior for (38) as shown in Fig. 7 is in accordancewith our previous observation from Fig. 5, and suggests thathigher-wavenumber instabilities also come into play. We note

30 W. Craig et al. / European Journal of Mechanics B/Fluids 32 (2012) 22–31

a

b

c

d

e

Fig. 7. Snapshots of the envelope magnitude |u| (thick solid line) and free-surfaceelevation ε−1η (thin solid line) for model (38) at (a) τ = 791.634, (b) 1399.943,(c) 1766.595, (d) 1924.922 and (e) 2500, for A0 = 0.15, ε = 1, ap = 0.01 andλp = 0.2.

interestingly that the profile of |u| does not exactly coincidewith the actual shape of the free-surface envelope everywhere atevery instant, although there is some correlation in the positionof their respective maximum amplitudes. This difference may be

explained in part by the fact that the relation (11) between u andη is not a simple relation of proportionality due to the presenceof the Fourier multiplier a−1(Dx), and therefore one should becareful about the physical interpretation of u. In contrast, thedependent variable in the originalmodels of Trulsen andDysthe [4]and Trulsen et al. [5] is more closely related to the free-surfaceenvelope.

Finally we point out that, for a more correct reconstructionof the free-surface elevation from e.g. (36), the inverse Fouriertransform F −1 in (46) should be translated byX → X − ∂kω(k0)τ/

√ε,

and multiplied by the additional phase factore−i[ω(k0)−k0∂kω(k0)]τ/ε,

which is related to the subtractions ofM and I from H as discussedin Section 5. Similar considerations apply to model (38). Theseadjustments however have no major effect on the results shownin Figs. 6 and 7.

9. Conclusions

We have applied the Hamiltonian approach of Craig et al. [8,9]to deriving Hamiltonian versions of the higher-order NLS modelsfor broader-banded deep-water waves, originally proposed byTrulsen and Dysthe [4] and Trulsen et al. [5]. The Benjamin–Feirinstability regions for these new models were then determined,and a good agreement was found in comparison with previouswork. Finally, numerical simulations were shown to illustratethese stability results and check the conservative properties ofour models. With this aim, we have introduced an efficient andaccurate symplectic scheme for time integration, combined witha pseudospectral method for space discretization.

Acknowledgments

W.C. is partially supported by the Canada Research ChairsProgram and NSERC through grant No. 238452-06. P.G. is partiallysupported by the National Science Foundation through grant No.DMS-0920850. C.S. is partially supported by NSERC through grantNo. 46179-05.

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