47

Fifth‐order nonlinear spectral model for surface gravity waves:
From pseudo‐spectral to spectral formulations
Wooyoung Choi
Department of Mathematical Sciences

New Jersey Institute of Technology
Newark, NJ 07102‐1982, USA

Abstract

We present a fifth‐order nonlinear spectral model describing the spectral evolution of
nonlinear surface gravity waves in water of finite depth. Using the equivalence between
pseudo‐spectral and spectral formulations, it is shown that the spectral model can be easily
obtained using a truncated Hamiltonian from the pseudo‐spectral formulation. The fifth‐

order model is written explicitly in terms of two canonical variables (the Fourier transformsof the surface elevation and the free surface velocity potential) and preserves the Hamiltonian
structure of the original water wave problem. Under discrete approximation, the time‐
periodic solutions of the spectral model for progressive and standing waves are shown to beconsistent with the classical solutions of Stokes and Rayleigh, respectively, when truncatedat the third order.

1

Introduction

For three‐dimensional water waves, the free surface boundary conditions can be written (Za‐kharov 1967), in terms of the surface elevation \zeta(x, t) and the free surface velocity potential\Phi(x, t) , as\frac{\partial\zeta}{\partial t}+\nabla\Phi\cdot\nabla\zeta=(1+ |\nabla\zeta|^{2})W, \frac{\partial\Phi}{\partial t}+\frac{1}{2}|\nabla\Phi|^{2} +g\zeta=\frac{1}{2}(1+|\nabla\zeta|^{2})W^{2},
where

x

(1.1)

is the the horizontal coordinate, t is time, \nabla is the horizontal gradient, and g is thegravitational acceleration. In (1.1), \Phi(x, t)\equiv\phi(x, z=\zeta, t) and W\equiv\partial\phi/\partial z(x, z=\zeta, t) are
the velocity potential and the vertical velocity evaluated at the free surface, respectively, where\phi and z are the three‐dimensional velocity potential and the vertical coordinate, respectively.
The two equations in (1.1) can be regarded as a system of nonlinear evolution equations for \zeta
and \Phi once W is expressed in terms of \zeta and \Phi . Depending upon how to close this system,
various theoretical models have been developed.
A theoretical model particularly useful for numerical computations of the evolution ofbroadband nonlinear surface waves was proposed by West et al.
RIMS Workshop on Nonlinear Water Waves, May 22‐26,
the occasion of his retirement

201S :(1987), who wroteWin

In honor of Professor Mitsuhiro Tanaka on

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an infinite series that depend on \zeta and \Phi . By substituting the infinite series into (1.1), a
closed system of nonlinear evolution equations for \zeta and \Phi was obtained. After assuming
the wave steepness is small, the series can be truncated at a desrired order of nonlinearity andthe resulting system has been studied numerically using a pseudo‐spectral method by numerous

researchers, including, for example, Tanaka (2001a, b) , Bateman et al. (2001), Choi et al (2005),
and Goullet & Choi (2011). Similar approaches have been proposed by Dommermuth & Yue(1987), Criag& Sulem (1993), and Clamond& Grue (2001).
An alternative approach to describe the evolution of boradband nonlinear waves was pro‐

posed by Zakharov (196S), who obtained a nonlinear integro‐differential equation in spectral
space for a single complex amplitude, which is a linear combination of the Fourier transforms of\zeta and \Phi . As a number of multiple integrals are required to be evaluated, the evolution equationof Zakharov is less efficient for numerical computations than the pseudo‐spectral model of Westet al. (1987). Nevertheless, his evolution equation is so useful for further analysis to describe
the time evolution of wave spectra. For example, in his seminal work, Zakharov (196S) reduced
the third‐order equation to a relatively simpler form for resonant four‐wave interactions. Thisequation is also often referred to as the (reduced) Zakharov equation, which has been studied
numerically (Annenkov & Shriar 2001). The spectral models of Zakharov (1968) have beenfurther extended to the fourth order by Stiassnie & Shemer (1984) to describe the five‐wave
interactions of gravity waves. Later the spectral models were reformulated by Krasitskii (1994)
directly from a Hamiltonian approach along with canonical transformations to simplify the
Hamiltonian. For the earlier development of of the spectral formulation, see, for example, Yuen

& Lake (1982) and Mei et al. (2005).
As one can imagine, the formulation of Zakharov (196S) should be equivalent to that ofWest et al. (1987). Therefore, it is expected to be straightforward to recover one formulationfrom the other. This is particularly useful if one is interested in a spectral model valid at ahigh order as the pseudo‐spectral model of West et al. (1987) can be found conveniently at anyorder of nonlinearity through recursion formulas. Here it is shown that a fifth‐order spectral
model can be indeed obtained in a straightforward manner from the pseudo‐spectral model ofWest et al. (1987) by taking advantage of its Hamiltonian structure.

2

Pseudo‐spectral formulation

2.1

Expansion

By expansing

W in Taylor series about

expression for

W can be written in infinite series as

z=0 ,it was shown by West et al. (1987) that the

W= \sum_{n=1}^{\infty}W_{n}, W_{n}= \sum_{j=0}^{n-1}C_{j}[\Phi_{n-j}]

for n\geq 1 ,(2.1)

where \Phi_{n} are given by\Phi_{1}=\Phi,\Phi_{n}=\sum_{j=1}^{n-1}A_{j}[\Phi_{n-j}]

for n\geq 2 ,

(2.2)

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and operators \mathcal{A}_{n} and C_{n} are defined, with \triangle=\nabla^{2} , by\mathcal{A}_{2m}=(-1)^{m+1}\frac{\zeta^{2m} {(2m)!}\triangle, \mathcal{A}_{2m+ 1}=(-1)^{m}\frac{\zeta^{2m+1} {(2m+1)!}\triangle^{m}\mathcal{L} ,C_{2m}=(-1)^{m+1} \frac{\zeta^{2m}}{(2m)!}\triangle^{m}\mathcal{L}, C_{2m+1}=(- 1)^{m+1}\frac{\zeta^{2m+1}}{(2m+1)!}\triangle^{m+1} .

(2.3)
(2.4)

\mathcal{L}[f]=\mathcal{F}^{-1}[-k\tanh(kh)\mathcal{F}[f]The linear operator \mathcal{L}[f] is given by, where h is the water depth,
\mat
h
cal
{
F
}
\mat
h
cal
{
F
}
^
{
1
}
and
and
represent the Fourier transform and its inverse, respectively. Alternatively, thelinear operator \mathcal{L} can be written as \mathcal{L}[f]=\int K(x-\xi)f(\xi)d\xi , where the kernel K(x) is definedin Fourier space as \mathcal{F}[K(x)]=-k\tanh(kh) .

Although the expansion for

W given by(2.1) requires no formal introduction of a small

parameter, except for the existence of Taylor series, the series given by (2.1)-(2.2) can beconsidered as an expansion in terms of (small) wave steepness, in particular, when the infinite
series need to be truncated for numerical simulations or further approximations. From \zeta\nabla=O(\epsilon) and \zeta \mathcal{L}=O(\epsilon) , where \epsilon=a/\lambda with a and \lambda being the characteristic wave amplitude andwavelength, respectively, one can see that \Phi_{n}=O(\epsilon^{n}) and W_{n}=O(\epsilon^{n}) . Therefore, the rate ofconvergence is expected to improve as \epsilon decreases.

2.2

System of West et al. (1987)

By substituting into (1.1) the expansion for W given by (2.1), the evolution equations for \zeta and\Phi are given by\frac{\partial\zeta}{\partial t}=\sum_{n=1}^{\infty}Q_{n}(\zeta, \Phi) , \frac {\partial\Phi}{\partial t}=\sum_{n=1}^{\infty}R_{n}(\zeta, \Phi)

,

(2.5)

where Q_{n} and R_{n} are given by
Q_{1}=W_{1},Q_{2}=W_{2}-\nabla\Phi\cdot\nabla\zeta,

Q_{n}=W_{n}+|\nabla\zeta|^{2}W_{n-2} for n\geq 3 ,(2.6)

R_{1}=-g\zeta, R_{2}=-\frac{1}{2}|\nabla\Phi|^{2}+\frac{1}{2}W_{1}^{2} R_{3}= W_{1}W_{2},R_{n}=\frac{1}{2}\sum_{j=0}^{n-2}W_{n-j 1}W_{j+1}+\frac{1}{2}|\nabla\zeta|^{2} \sum_{\dot{j}=0}^{n-4}W_{n-j 3}W_{j+1}

for n\geq 4 .

(2.7)

Here the expressions of W_{n} are given by (2.1). Notice that Q_{n}=O(\epsilon^{n}) are linear in \Phi while
R_{n}=O(\epsilon^{n}) are quadratic in \Phi.
For small amplitude waves, the system (2.5) can be linearized, with W_{1}=-\mathcal{L}[\Phi] , to\frac{\partial\zeta}{\partial t}=-\mathcal{L}[\Phi], \frac{\partial\Phi} {\partial t}=-g\zeta ,(2.8)

which can be combined into \partial^{2}\zeta/\partial t^{2}=g\mathcal{L}[\zeta] . The same equation also holds for \Phi . Substituting(\zeta, \Phi)\sim\exp[i(k\cdot x-\omega t)] into (2.8) yields the linear dispersion relation given by\omega^{2}=gk\tanh kh ,

(2.9)

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where we have used\mathcal{L}[e^{i}k\cdot x]=-k\tanh kh e^{ik\cdot x} . While the leading‐order terms ( Q_{1} and R_{1} )represent linear dispersive effects, Q_{n} and R_{n} for n\geq 2 describe nonlinear dispersive effects
and nonlinear wave interactions.

Following West et al. (1987), the system given by (2.5) has been studied extensively
in recent years using a pseudo‐spectral method based on Fast Fourier Transform (FFT), forexample, by Tananka (2001a, b) and many others. For numerical computations, after assuming\zeta and \Phi are doubly periodic in space so that they can be written in Fourier series, the linear
operators \triangle and \mathcal{L} in (2.3)-(2.4) are evaluated in Fourier space:\triangle=-k_{j}^{2}, \mathcal{L}=-k_{j}T_{j} ,(2.10)

where j=(j, l), k_{j}=(jK_{x}, lK_{y}), k_{j}=|k_{j}|, T_{j}=\tanh(k_{j}h) , and with K_{x} and K_{y} being thefundamental wavenumbers in the x and y directions, respectively. Then the two nonlinear
operators \mathcal{A}_{n} and C_{n} defined by (2.3) and (2.4) are computed as\mathcal{A}_{2m}=-\frac{\zeta^{2m}}{(2m)!}k_{j}^{2m} \mathcal{A}_{2m+1}=-\frac {\zeta^{2m+1}}{(2m+1)!}k_{j}^{2m+1}T_{j} ,C_{2m}= \frac{\zeta^{2m}}{(2m)!}k_{j}^{2m+1}T_{j}, C_{2m+1}=\frac{\zeta^{2m+1}} {(2m+1)!}k_{j}^{2m+2}(2.11)(2.12)In (2.11), to compute \mathcal{A}_{2m}[f] , the Fourier transform of f is multiplied by -k_{j}^{2m}/(2m)! in Fourier

space and, then, its inverse Fourier transform is multiplied by \zeta^{2m} in physical space. Finally,
after evaluating its right‐hand sides up to a desired order of nonlinearity, the system given by(2.5) is integrated in time.
2.3

Hamiltonians

Zakharov (196S) showed that the totoal energy defined by

E= \frac{1}{2}\int(g\zeta^{2}+\Phi\frac{\partial\zeta}{\partial t}) d ,(2.13)xis the Hamiltonian for the water wave problem so that the evolution equations for \zeta and \Phi canbe written as\frac{\partial\zeta}{\partialt}=\frac{\deltaE}{\delta\Phi},\frac{0\Phi} {\partialt}=-\frac{\deltaE}{\delta\zeta} ,(2.14)

where \delta E/\delta\zeta and \delta E/\delta\Phi represent the functional derivatives of E with respect to the two
conjugate variables \zeta and \Phi , respectively. Therefore, the total energy E is conserved. From

(2.5), the total energy E defined in (2.13) can be expanded, in infinite series, as

E= \frac{1}{2}\int(g\zeta^{2}+\Phi\sum_{n=1}^{\infty}Q_{n})dx=\sum_{n=2} ^{\infty}E_{n},(2.15)

where Q_{n} are given by (2.7) and the n‐th order energy E_{n} is given by

E_{2}= \frac{1}{2}\int(g\zeta^{2}+\Phi Q_{1}) dx,

E_{n}= \frac{1}{2}\int\Phi Q_{n-i}dx for n) 3.

(2.16)

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2.4

Fifth‐order model

When truncated at

O(\epsilon^{5}) , the fifth‐order nonlinear evolution equations for \zeta and \Phi can beobtained as\frac{\partial\zeta}{\partial t}=\sum_{n=1}^{5}Q_{n}(\zeta, \Phi) , \frac{\partial\Phi}{\partial t}=\sum_{n=1}^{5}R_{n}(\zeta, \Phi),(2.17)

where Q_{n} and R_{n} are given, explicitly, by

Q_{1}=-\mathcal{L}[\Phi] ,(2.18)(2.19)
Q_{2}=-\nabla\cdot(\zeta\nabla\Phi)-\mathcal{L}[\zeta \mathcal{L}[\Phi] ,(2.20)
Q_{3}=- \mathcal{L}[\zeta \mathcal{L}[\zeta \mathcal{L}[\Phi] +\frac{1}{2} \zeta^{2}\nabla^{2}\Phi]-\nabla^{2}(\frac{1}{2}\zeta^{2}\mathcal{L}[\Phi]) ,Q_{4}=- \mathcal{L}[\zeta \mathcal{L}[\zeta \mathcal{L}[\zeta \mathcal{L}[\Phi] ]+\frac{1}{2}\zeta^{2}\nabla^{2}\Phi]+\frac{1}{2}\zeta^{2}\nabla^{2}(\zeta \mathcal{L}[\Phi])-\frac{1}{6}\zeta^{3}\nabla^{2}\mathcal{L}[\Phi]- \nabla^{2}(\frac{1}{2}\zeta^{2}\mathcal{L}[\zeta \mathcal{L}[\Phi] +\frac{1} {3}\zeta^{3}\nabla^{2}\Phi) , (2.21)
Q_{5}=- \mathcal{L}[\zeta \mathcal{L}[\zeta \mathcal{L}[\zeta \mathcal{L}[\zeta \mathcal{L}[\Phi] +\frac{1}{2}\zeta^{2}\nabla^{2}\Phi]+\frac{1}{2}\zeta^{2} \nabla^{2}(\zeta \mathcal{L}[\Phi])-\frac{1}{6}\zeta^{3}\nabla^{2}\mathcal{L} [\Phi]+ \frac{1}{2}\zeta^{2}\nabla^{2}(\zeta \mathcal{L}[\zeta \mathcal{L}[\Phi] + \frac{1}{2}\zeta^{2}\nabla^{2}\Phi)-\frac{1}{6}\zeta^{3}\nabla^{2}\mathcal{L} [\zeta \mathcal{L}[\Phi] -\frac{1}{24}\zeta^{4}\nabla^{2}\nabla^{2}\Phi]- \nabla^{2}(\frac{1}{2}\zeta^{2}\mathcal{L}[\zeta \mathcal{L}[\zeta \mathcal{L}[\Phi] +\frac{1}{2}\zeta^{2}\nabla^{2}\Phi]+\frac{1}{3}\zeta^{3} \nabla^{2}(\zeta \mathcal{L}[\Phi])-\frac{1}{8}\zeta^{4}\nabla^{2}\mathcal{L} [\Phi]) , (2.22)
R_{1}=-g\zeta ,(2.23)

R_{2}=- \frac{1}{2}\nabla\Phi\cdot\nabla\Phi+\frac{1}{2}(\mathcal{L}[\Phi])^{2} ,(2.24)

R_{3}=\mathcal{L}[\Phi](\mathcal{L}[\zeta \mathcal{L}[\Phi] +\zeta\nabla^{2} \Phi) ,(2.25)

R_{4}= \mathcal{L}[\Phi]\mathcal{L}[\zeta \mathcal{L}[\zeta \mathcal{L}[\Phi] + \frac{1}{2}\zeta^{2}\nabla^{2}\Phi]+\nabla^{2}[\frac{1}{4}\zeta^{2}(\mathcal{L}[ \Phi])^{2}]+\frac{1}{2}(\mathcal{L}[\zeta \mathcal{L}[\Phi] +\zeta\nabla^{2} \Phi)^{2}+ \frac{1}{2}\zeta(\nabla^{2}\zeta)(\mathcal{L}[\Phi])^{2}-\frac{1}{2}\zeta^{2} (\nabla \mathcal{L}[\Phi])^{2}(2.26)

R_{5}= \mathcal{L}[\Phi]\mathcal{L}[\zeta \mathcal{L}[\zeta \mathcal{L}[\zeta \mathcal{L}[\Phi] +\frac{1}{2}\zeta^{2}\nabla^{2}\Phi]+\frac{1}{2}\zeta^{2} \nabla^{2}(\zeta \mathcal{L}[\Phi])-\frac{1}{6}\zeta^{3}\nabla^{2}\mathcal{L} [\Phi]+ \nabla^{2}(\frac{1}{2}\zeta^{2}\mathcal{L}[\Phi]\mathcal{L}[\zeta \mathcal{L} [\Phi] +\frac{1}{3}\zeta^{3}\mathcal{L}[\Phi]\nabla^{2}\Phi)-\zeta^{2}(\nabla \mathcal{L}[\Phi])\cdot(\nabla \mathcal{L}[\zeta \mathcal{L}[\Phi] +\frac{2}{3} \zeta\nabla(\nabla^{2}\Phi)+ \frac{1}{6}\zeta^{3}(\nabla^{2}\mathcal{L}[\Phi])(\nabla^{2}\Phi)+ (\mathcal{L}[\zeta \mathcal{L}[\Phi] +\zeta\nabla^{2}\Phi)(\mathcal{L}[\zeta \mathcal{L}[\zeta \mathcal{L}[\Phi] +\frac{1}{2}\zeta^{2}\nabla^{2}\Phi]+ \frac{1}{2}\zeta(\nabla^{2}\zeta)\mathcal{L}[\Phi])(2.27)The expressions of the corresponding Hamiltonians E_{n}(n=2, \cdots, 6) are explicitly given by

E_{2}= \frac{1}{2}\int(g\zeta^{2}-\Phi \mathcal{L}[\Phi]) dx,
E_{3}= \frac{1}{2}\int\{\zeta\nabla\Phi\cdot\nabla\Phi-\zeta(\mathcal{L}[\Phi]) ^{2}\} dx,
E_{4}=- \frac{1}{2}\int\zeta \mathcal{L}[\Phi](\mathcal{L}[\zeta \mathcal{L} [\Phi] +\zeta\nabla^{2}\Phi) dx,(2.28)(2.29)(2.30)

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E_{5}=- \frac{1}{2}\int\{\zeta(\mathcal{L}[\zeta \mathcal{L}[\Phi] )^{2}+ \frac{1}{3}\zeta^{3}(\nabla^{2}\Phi)^{2}+ \zeta \mathcal{L}[\Phi](\mathcal{L}[\zeta^{2}\nabla^{2}\Phi]+\frac{1}{2}\zeta \nabla^{2}(\zeta \mathcal{L}[\Phi])-\frac{1}{6}\zeta^{2}\nabla^{2}\mathcal{L} [\Phi])\} dx ,(2.31)

E_{6}=- \frac{1}{2}\int\{\zeta \mathcal{L}[\zeta \mathcal{L}[\zeta \mathcal{L}[ \Phi] (\mathcal{L}[\zeta \mathcal{L}[\Phi] +\zeta\nabla^{2}\Phi)+\mathcal{L} [\zeta \mathcal{L}[\Phi] (\zeta^{2}\nabla^{2}(\zeta \mathcal{L}[\Phi])-\frac{1} {3}\zeta^{3}\nabla^{2}\mathcal{L}[\Phi])+( \frac{1}{2}\zeta^{2}\nabla^{2}\Phi)\mathcal{L}[\frac{1}{2}\zeta^{2} \nabla^{2}\Phi]+\zeta^{2}\mathcal{L}[\Phi](\frac{1}{2}\nabla^{2}(\zeta^{2} \nabla^{2}\Phi)-\frac{1}{12}\zeta^{2}\nabla^{2}(\nabla^{2}\Phi) \} dx . (2.32)

When truncated at O(\epsilon^{3}) , the system given by (2.17) becomes the third‐order system obtainedby Choi (1995), who showed that the truncated system also preserves the Hamiltonian structure.
Likewise, it can be shown that the fifth‐order model given by (2.17) is a Hamiltonian system.

3

Spectral Formulation

3.1

System for continuous spectrum

To obtain a nonlinear system in spectral space, \zeta and \Phi are expressed as\zeta(x, t)=\int a(k, t)e^{-ik\cdot x} d , \Phi(x, t)=\int b(k, t)e^{-ik\cdot x} d ,
k

k

(3.1)

where a(k, t) and b(k, t) representing the Fourier transforms of \zeta and \Phi , respectively. Noticethat a(-k, t)=a^{*}(k, t) and b(-k, t)=b^{*}(k, t) , with the asterisks representing the complex
conjugates, as \zeta and \Phi are real‐valued functions.

One way to find such a system is to take the Fourier transform of (2.5), which would yield
the nonlinear evolution equations for a(k, t) and b(k, t) as\frac{\partial a}{\partial t}-kTb=\sum_{n=2}^{\infty}q_{n}, \frac{\partial b}{ \partial t}+ga=\sum_{n=2}^{\infty}r_{n}

,

(3.2)

where q_{n} and r_{n} representing the Fourier transforms of Q_{n} and R_{n} given by (2.6)-(2.7) can be
written asq_{n}= \iint\cdot\cdot\int\alpha_{0,1,\cdots,n}^{(n)}b_{1}a_{2}a_{3}\cdots a_{n}\delta_{0-1-n}dk_{1}dk_{2}\cdots dk_{n} ,r_{n}= \iint\cdot\int\beta_{0,1,\cdots,n}^{(n)}b_{1}b_{2}a_{3}\cdots a_{n} \delta_{0-1-n}dk_{1}dk_{2}\cdots dk_{n} .

(3.3)

(3.4)

In (3.3)-(3.4) , we have used the following short‐hand notationsa_{j}=a(k_{j}, t) , b_{j}=b(k_{j}, t) , \delta_{j-l}=\delta(k_{j}-k_{l}) , k_{0}=k ,(3.5)

where \delta is the Dirac delta function. Under the third‐order approximation, the system given by(3.2) was derived first by Zakharov (196S) for infinitely deep water and by Stiassnie & Shemer(1984) for finite depth water. The system has been also extended to O(\epsilon^{4}) by Stiassnie &Shemer (1984).
Although it is straightforward, finding the explicit expressions of\alpha_{0,1,\cdots,n}^{(n)} and \beta_{0,1,\cdots,n}^{(n)} bytaking the Fourier transform of (2.5) is lengthy and cumbersome, in particular, as the order

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of nonlinearity increases. An alternative and more convenient way is to use the Hamiltonian,as shown by Krasitskii (1994), whose approach will be adopted here to obtain the fifth‐ordersystem. From (2.16), the n‐th order Hamiltonian H_{n}=E_{n}/(2\pi)^{2} can be written in spectral
space asH_{2}= \frac{1}{2}\i nt(ga_{1}a_{2}+k_{1}T_{1}b_{1}b_{2})\delta_{1+2}dk_{1} dk_{2} ,H_{n}= \frac{1}{2}\i nt\cdot\cdot\int h_{1,2,3,\cdots,n}^{(n)}b_{1}b_{2}a_{3} a_{n}\delta_{1+\cdots+n}dk_{1}dk_{2}dk_{3}\cdots dk_{n} forFor example, under the fifth‐order approximation,
explicitly as(3.6)
n\geq 3 .

(3.7)

h_{1,2,3,\cdots,n}^{(n)} for n=3,4,5,6 can be written

h_{1,2,3}^{(3)}=-(k_{1}\cdot k_{2}+\theta_{1}\theta_{2}) ,(3.8)

h_{1,2,3,4}^{(4)}=-(k_{2}^{2}\theta_{1}-\theta_{1}\theta_{2}\theta_{2+3}) ,h_{1,2,3,4,5}^{(5)}=-[( \frac{1}{6}k_{2}^{2}-\frac{1}{2}k_{2+3}^{2}+\theta_{1+ 3}\theta_{2+4})\theta_{1}\theta_{2}-k_{2}^{2}\theta_{1}\theta_{2+3+4}+\frac{1} {3}k_{1}^{2}k_{2}^{2}]
h_{1,2,3,4,5,6}^{(6)}=[( \theta_{1}\theta_{1+3}-k_{1}^{2})\theta_{2}\theta_{2+ 4}\theta_{2+4+5}+(\frac{1}{3}k_{2}^{2}-k_{2+4}^{2})\theta_{1}\theta_{2}\theta_{1 +3}+ \frac{1}{4}k_{1}^{2}k_{2}^{2}\theta_{2+5+6}+\frac{1}{2}k_{1}^{2}k_{1+3+4}^{2} \theta_{2}-\frac{1}{12}k_{1}^{4}\theta_{2}](3.9)(3.10)(3.11)

wherek_{j}=|k_{j}|,\theta_{j}=k_{j}T_{j},T_{j}=\tanh(k_{\dot{j}}h),k_{m+n}=|k_{m}+k_{n}|,T_{m+n}=\tanh(k_{m+n}h)(3.12).

The evolution equations for a(k, t) and b(k, t) can be then obtained from Hamilton’s equations:\frac{\partiala}{\partialt}=\frac{\deltaH}{\deltab^{*} ,\frac{\partialb} {\partialt}=-\frac{\deltaH}{\delta ^{*} .

(3.13)

From (3.6)-(3.7) and (3.13), the expressions of \alpha_{0,1,\cdots,n}^{(n)} and \beta_{0,1\cdots,n}^{(n)} in (3.3)-(3.4) are found, in
terms of

h_{1,2,3,\cdots,n}^{(n)} , as\alpha_{0,1,\cdots,n}^{(n)}=\frac{1}{2}(h_{-0,1,2,\cdots,n}^{(n+1)}+h_{1,-0,2, \cdots,n}^{(n+1)}) ,\beta_{0,1\cdots,n}^{(n)}=-\frac{1}{2} (h_{1,2,-0,3,\cdots,n}^{(n+1)}+ +h_{1, 2,3,\cdots,n,-0}^{(n+1)})
Notice that the interaction coefficientswords, except for

h_{1,2,3}^{(3)}(3.14)(3.15)

h_{1,2,\cdots,n}^{(n)} in (3.8)-(3.11) are not symmetric. In other, they change when indices 1 and 2 are interchanged although theirnHamiltonians given by (3.7) remain unchanged. This is also true for indices 3,. Never‐theless, if necessary, they can be easily made symmetric, as shown by Krasitskii (1994).

3.2

System for discrete spectrum

When a nonlinear wave field can be represented by a superposition of discrete modes, a(k, t)andb(k, t) can be written asa(k, t)= \sum_{j}\delta(k-k_{j})a_{j}(t) , b(k, t)=\sum_{j}\delta(k-k_{j})b_{j} (t) ,(3.16)

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where k_{-j}=-k_{j} . In (3.16), the summations should be in general taken over all discrete modes
involved in nonlinear wave interactions unless an additional approximation is made.
When truncated at O(\epsilon^{M}) , the amplitude equations for a_{j} and b_{j} under the M‐th orderapproximation are given, from (3.2), by\frac{da_{j} {dt}-k_{j}T_{j}b_{j}=\sum_{n=2}^{M}[\sum_{j_{1},j_{2},\cdot\cdot j_{n} .,\alpha_{j, _{1},\cdots,j_{n} ^{(n)}b_{j_{1} a_{j_{2} a_{j_{3} \cdots a_{j_{n} \delta_{0-1 n]}\frac{db_{j} {dt}+ga_{j}=\sum_{n=2}^{M}[\sum_{j_{1},j_{2},\cdot\cdot j_{n} ., \beta_{j, _{1},\cdots,j_{n} ^{(n)}b_{j_{1} b_{j_{2} a_{j_{3} \cdots a_{j_{n} \delta_{0-1 n]}(3.17)(3.18)

where \delta_{0-1-n}=\delta_{j-j_{1}-j_{n}} . The corresponding Hamiltonians are given by

H_{2}= \frac{1}{2}\sum_{j_{1} \sum_{j_{2} (ga_{j_{1} a_{j_{2} +k_{j_{1} T_{j_{1} b_{j_{1} b_{j_{2} )\delta_{1+2}
H_{n}= \frac{1}{2}\sum_{j_{1},j_{2},\cdot\cdot,j_{n} .h_{j_{1},j_{2} ^{(n)}, \cdot\cdot\cdot, j_{n}b_{j_{1} b_{j_{2} a_{j_{3} a_{j_{n} \delta_{1+2+\cdots+n},(3.19),(3.20)

from which the amplitude equations given by (3.17)-(3.18) can be obtained from the Hamilton’s
equations:\frac{\partial _{j}{\partialt}=\frac{\partialH}{\partialb_{j}^{*},\frac {\partialb_{j}{\partialt}=-\frac{\partialH}{\partial _{j}^{*}
where H= \sum_{n}H_{n} . For\alpha_{j, _{1} ^{(n\prime},\cdot\cdot\cdot,

j_{n}

and

\beta_{j, _{1} ^{(n\prime},\cdot\cdot\cdot,

j_{n}M=5 , the expressions of

h_{j_{1},j_{2} ^{(n)},,(3.21)j_{n}are defined by (3.8)-(3.11) andare given by (3.14)-(3.15) .

When k_{j}=(jK_{x}, lK_{y}), (3.16) represent the Fourier series of \zeta and \Phi and equations (3.17)‐(3.18) determine their evolution of the Fourier coefficients, a_{j} and b_{j} . If a finite number ofFourier modes are used for numerical computations, solving the ordinary differential equations

given by (3.17)-(3.18) is equivalent to solving (2.5) using the pseudo‐spectral method describedin §2.2. Unfortunately, the evaluation of the right‐hand sides is computationally expensive
and, therefore, solving a dynamical system is in general less effective than the pseudo‐spectral
method based on FFT.

3.3

Time‐periodic solutions of the third‐order spectral model

Under the third‐order approximation (M=3) , the amplitude equations for a_{j} and b_{j} can be
written, from (3. 17)-(3.18) , as\frac{da_{j} {dt}=k_{j}T_{j}b_{j}+\sum_{j_{1\dot{J}2} (k_{j}\cdot k_{j_{1} -k_ {j}T_{j}k_{j_{1} T_{j_{1} )b_{j_{1} a_{j_{2} \delta_{0-1-2}+ \sum_{j_{1\dot{J}2},j_{3} [k_{j}T_{j}(k_{j_{1} T_{j_{1} k_{j_{1}+j_{2} T_{j_{1}+j_{2} -\frac{1}{2}k_{j_{1} ^{2})-\frac{1}{2}k_{j}^{2}k_{j_{1} T_{j_{1} ]b_{j_{1} a_{j_{2} a_{j_{3} \delta_{0-1-2-3} , (3.22)\frac{db_{j} {dt}=-ga_{j}+\sum_{j_{1},j_{2} \frac{1}{2}(k_{j_{1} \cdot k_{j_{2} +k_{j_{1} T_{j_{1} k_{j_{2} T_{j_{2} )b_{j_{1} b_{j_{2} \delta_{0-1-2}

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RIMS Workshop on Nonlinear Water Waves+ \sum_{j_{1\dot{J}2}j_{3} [k_{j_{1} T_{j_{1} (-k_{j_{2} T_{j_{2} k_{j-j_{1} T_ {j-j_{1} +k_{j_{2} ^{2})]b_{j_{1} b_{j_{2} a_{j_{3} \delta_{0-1-2-3} .

(3.23)

When we assume that the waves are propagating in the x ‐direction so that k_{j}=(k_{j}, 0)with k_{j}=jk and (a_{j}, b_{j})=(a_{j}, b_{j}) , equations (3.22)-(3.23) describe the evolution of the Fourier
coefficients of \zeta and \Phi . Furthermore, we assume that the first harmonics are initially dominant
and all other higher‐harmonics are excited through nonlinearity so thata_{j}=O(\epsilon) , a_{0}=O(b_{0})=O(b_{2_{\dot{J}}})=O(a_{2j})=O(\epsilon^{2}) , a_{3j}=O(b_{3j})=O(\epsilon^{3}) .

(3.24)

Then, the third‐order system (3.22)-(3.23) can be approximated by four ordinary differential
equations: for the j‐th mode,\frac{da_{j} {dt}-k_{\dot{j} T_{j}b_{j}=2k_{j}^{2}(1-T_{j}T_{2_{\dot{J} }) a_{j}^{*}b_{2_{J} -k_{j}^{2}(1+T_{j}^{2})a_{2_{\dot{J} }b_{j}^{*}-2k_{j}^{3}T_{j}(1-T_{\dot{j}}T_{2_{\dot{J}}})|a_{j}|^{2}b_{j}-k_{j}^{3}T_{j}a_ {j}^{2}b_{j}^{*} ,\frac{db_{j} {dt}+ga_{j}=-2k_{\dot{j} ^{2}(1-T_{j}T_{2j})b_{j}^{*} b_{2_{\dot{j} }+2k_{\dot{j} ^{3}T_{j}(1-T_{j}T_{2j})a_{j}|b_{j}|^{2}+k_{\dot{j} ^{3}T_{j}a_{j}^{*}b_{\dot{j} ^{2} ,(3.25)(3.26)

and, for the 2j ‐th mode,\frac{da_{2j} {dt}-k_{2j}T_{2j}b_{2j}=2k_{j}^{2}(1-T_{j}T_{2_{\dot{j} })a_{j} b_{j} ,\frac{db_{2_{j} }{dt}+ga_{2_{j} =\frac{1}{2}k_{j}^{2}(1+T_{j}^{2})b_{j}^{2} .

(3.27)(3.28)The Hamiltonian for the system is given, by imposing (3.24) to (3.19)-(3.20) , by

H=(g|a_{j}|^{2}+k_{j}T_{j}|b_{j}|^{2})+(g|a_{2j}|^{2}+k_{2j}T_{2j}|b_{2j}|^{2})+ \frac{1}{2}[h_{j,\dot{j},-2_{\dot{j} }^{(3)}b_{j}^{2}a_{2j}^{*}+h_{-J,-j,2j}^ {(3)}b_{j}^{*2}a_{2j}+2h_{2_{\dot{j} ,-j,-j}^{(3)}b_{2j}b_{j}^{*}a_{j}^{*}+2h_{- 2j,j,j}^{(3)}b_{2j}^{*}b_{j}a_{j}]+ \frac{1}{2}[h_{j,\dot{j},-j,-j}^{(4)}b_{j}^{2}a_{j}^{*2}+h_{-\dot{j},-j,j,j}^ {(4)}b_{j}^{*2}a_{\dot{j} ^{2}+(h_{\dot{j},-j,j,-j}^{(4)}+h_{\dot{j},-j,-j,j}^{(4)}+h_{-j,j,j,-j}^{(4)}+h_{- j,j,-j,j}^{(4)})|b_{j}|^{2}|a_{j}|^{2}]=(g|a_{j}|^{2}+k_{j}T_{j}|b_{j}|^{2})+(g|a_{2j}|^{2}+k_{2_{\dot{j} } T_{2_{\dot{J} }|b_{2_{\dot{j} }|^{2})\frac{1}{2}k_{j}^{2}(1+T_{j}^{2})(b_{j}^{2}a_{2j}^{*}+b_{j}^{*2}a_{2_{\dot{j} })+2k_{j}^{2}(1-T_{j}T_{2j})(b_{2j}b_{j}^{*}a_{j}^{*}+b_{2j}^{*}b_{j}a_{j})‐ \frac{1}{2}k_{j}^{3}T_{j}(b_{j}^{2}a_{j}^{*2}+b_{\dot{j} ^{*2}a_{\dot{j} ^{2}) -2k_{\dot{j} ^{3}T_{j}(1-T_{j}T_{2j})|b_{j}|^{2}|a_{j}|^{2}‐

where we have used

(3.29)

h_{2j,-j,-j}^{(3)}=h_{-j,2j,-j}^{(3)} and h_{-2j,j,j}^{(3)}=h_{j,-2_{\dot{J} ,j}^{(3)} . Here the amplitude equationsof the third‐harmonics ( a_{3_{j}} and b_{3_{\dot{j} } ) as not written as they have no effect on the dynamics ofthe first harmonics of interest unless the higher‐order nonlinearity is included. Therefore, we
consider only the first two harmonics here.

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3.3.1

Progressive waves

When linearized, (3.25)-(3.26) can be reduced to\frac{da_{j} {dt}=k_{\dot{j} T_{j}b_{\dot{j} , \frac{db_{j} {dt}=-ga_{j} ,(3.30)

whose solution can be written asa_{j}=\overline{a}_{j}e^{i\omega_{j}t} b_{\dot{j}}=i(g/\omega_{j})\overline{a}_ {j}e^{i\omega_{j}t}(3.31)

b_{j}=i(g/\omega_{j})a_{j} .

(3.32)

so thatHere \omega_{j}>0 satisfies the linear dispersion relation (2.9):\omega_{\dot{j} ^{2}=gk_{j}T_{j} .

(3.33)At the second order, the particular solutions of (3.27)-(3.28) for the second harmonics can beobtained, using da_{2j}/dt=2\omega_{j}a_{2j} and db_{2_{\dot{J}}}/dt=2\omega_{j}b_{2_{\dot{J}}} , asa_{2j}= \frac{1}{\omega_{2j}^{2}-4\omega_{\dot{j} ^{2} [4i\omega_{j}k_{j}^{2}(1 -T_{j}T_{2j})a_{j}b_{\dot{j} +k_{j}^{3}T_{2j}(1+T_{j}^{2})b_{\dot{j} ^{2}]b_{2j}= \frac{1}{2_{\dot{j} ^{-4\omega_{j}^{2} 2}[-2gk_{j}^{2}(1-T_{j}T_{2j})a_ {j}b_{j}+i\omega_{\dot{j} k_{\dot{j} ^{2}(1+T_{j}^{2})b_{\dot{j} ^{2}](3.34)(3.35)

where \omega_{2j}^{2}=gk_{2j}T_{2j} is the natural frequency of the second harmonics of wavenumber k_{2j}=2k_{j}and we have assumed that \omega_{2j}\neq 2\omega_{j} . When substituting the linear solution (3.31) into (3.34)‐(3.35), the second harmonic solutions can be found asa_{2j}= \alpha_{2j}\overline{a}_{j}^{2}e^{2i\omega_{j}t \alpha_{2j}=gk_{j}^{2} (\frac{4-3T_{j}T_{2j}+T_{2j}/T_{j} {4\omega_{\dot{j} ^{2}-\omega_{2j}^{2} )= k_{j}(\frac{3-T_{\dot{j} ^{2} {2T_{j}^{3} )b_{2j}= i\beta_{2j}\overline{a}_{\dot{j} ^{2}e^{2i\omega_{j}t \beta_{2j}=\frac {g^{2}k_{j}^{2} {\omega_{j} (\frac{3-2T_{j}T_{2j}+T_{j}^{2} {4\omega_{j}^{2}- \omega_{2j}^{2} )=\omega_{j}(\frac{3+T_{j}^{4} {4T_{\dot{j} ^{4} )(3.36)(3.37)

where we have used, for the last expressions of \alpha_{2j} and \beta_{2_{\dot{J} },T_{2j}=2T_{j}/(1+T_{\dot{j}}^{2}) .

(3.38)

From (3.31), a_{2j} and b_{2j} given by (3.36)-(3.37) can be expressed, in terms of a_{j} and b_{j} , asa_{2_{\dot{J}}}=\alpha_{2j}a_{\dot{j}}^{2}+O(\epsilon^{3}) , b_{2j}=i\beta_{2j} a_{\dot{j}}^{2}+O(\epsilon^{3}) .

(3.39)To study the nonlinear behavior of the first harmonics ( a_{j} and b_{j} ), although not necessary,it is convenient to use a single amplitude equation, for example, for a_{j} . After substituting (3.32)
and (3.39) into the right‐hand sides of (3.25)-(3.26) , the time evolution equation for a_{j} correcttoO(\epsilon^{3}) can be found as\frac{d^{2}a_{j} {dt^{2} +\omega_{j}^{2}(1+\alpha_{j}|a_{\dot{j} |^{2})a_{j}=0 ,(3.40)

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where \alpha_{j} is given by\alpha_{j}=k_{j}^{2}[\frac{16T_{j}+(1-18T_{j}^{2}+9T_{j}^{4})T_{2j} {2T_{j} (2T_{\dot{j} -T_{2j})}]=k_{j}^{2}(\frac{9T_{j}^{4}-10T_{j}^{2}+9}{2T_{j}^{3} )>0.

(3.41)

Similarly, the amplitude equation for b_{j} can be found as\frac{d^{2}b_{j} {dt^{2} +\omega_{\dot{j} ^{2}(1+\beta_{j}|b_{j}|^{2})b_{j}=0, \beta_{j}=(\omega_{j}^{2}/g^{2})\alpha_{j} .

(3.42)For a time‐periodic solution of (3.40), a_{j}(t) is written asa_{j}(t)=A_{j}e^{i\Omega_{j}t} \Omega_{j}=\omega_{j}[1+\delta_{j}+ O(\epsilon^{4})](3.43)

where \delta=O(\epsilon^{2}) is the nonlinear correction to the wave frequency. By substituting (3.43) into
(3.40), one can find, at the order of O(\epsilon^{3}) , that\delta_{j}=\frac{1}{2}\alpha_{j}A_{j}^{2}=(\frac{9T_{j}^{4}-10T_{j}^{2}+9}{4T_ {j}^{3} )k_{j}^{2}A_{j}^{2},(3.44)

which is the nonlinear frequency correction of Stokes waves in water of finite depth, as shown,for example, in Whitham (1976, §13.13). As a special case, for infinitely deep water (T_{j}arrow 1
and T_{2_{j}}arrow 1) , the expression of \alpha_{j}, \delta_{\dot{j} , and a_{2_{\dot{J} } are given by\alpha_{j}=4k_{j}^{2}, \delta_{j}=2k_{j}^{2}A_{j}^{2}, a_{2j}=k_{j}A_{j}^{2} e^{2i\omega_{j}t}(3.45)

This solution corresponds to that of Stokes (1847), where the wave amplitude \overline{a}_{j} is defined as\overline{a}_{j}=2A_{j} so that \delta_{j}=k_{\dot{j} ^{2}\overline{a}_{\dot{j} ^{2}/2.
3.3.2

Standing waves

For standing wave solutions, we must havea_{j}=a_{-j}=a_{j}^{*}, b_{j}=b_{-j}=b_{j}^{*} , a_{0}=0 ,(3.46)

so that \zeta and \Phi can be written as\zeta(x, t)=2\sum_{j\geq 0}a_{j}(t)\cos(k_{j}x), \Phi(x, t)=2\sum_{j\geq 0} b_{j}(t)\cos(k_{j}x) ,which satisfy the side‐wall boundary conditions at(3.47)

x=0 and L withk_{j}=j\pi/L ,(3.48)

where L is the tank length. Then, as a_{j} and b_{j} are real, their evolution equations are given,from (3.22)-(3.23) , or directly from (3.25)-(3.26) , by\frac{da_{j} {dt}-k_{j}T_{j}b_{\dot{j} =2k_{j}^{2}(1-T_{\dot{j} T_{2j})a_{\dot {j} b_{2_{\dot{j} }-k_{j}^{2}(1+T_{j}^{2})a_{2_{\dot{j} }b_{j}-k_{j}^{3}T_{j}(3- 2T_{j}T_{2j})a_{\dot{j} ^{2}b_{j} , (3.49)(3.50)\frac{db_{j} {dt}+ga_{j}=-2k_{j}^{2}(1-T_{\dot{j} T_{2_{\dot{j} })b_{j}b_{2j}+ k_{\dot{j} ^{3}T_{\dot{j} (3-2T_{j}T_{2j})a_{\dot{j} b_{\dot{j} ^{2} ,58
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while the time evolution of the second harmonics a_{2j} and b_{2j} are governed by (3.27)-(3.28) .
When we linearize (3.49)-(3.50) , the leading‐order solutions can be found asa_{j}=A_{j}e^{i\omega_{j}t}+C.C . ,b_{j}=i(g/\omega_{j})A_{j}e^{i\omega_{j}t}+C.C . ,(3.51)

where the complex conjugates (C.C.) are needed as a_{j} and b_{j} are real functions. At the second‐order, the particular solutions of (3.27)-(3.28) for the second harmonics can be found asa_{2_{J} =(\alpha_{2_{\dot{J} }A_{j}^{2}e^{2i\omega_{j}t}+C.C.)+\gamma_{2_{\dot {J} }|A_{j}|^{2} b_{2j}=(i\beta_{2j}A_{\dot{j} ^{2}e^{2i\omega_{j}t}+C.C.) ,(3.52)

with \alpha_{2j}, \beta_{2_{\dot{j} , and \gamma_{2j} given by\alpha_{2j}=k_{j}(\frac{3-T_{j}^{2} {2T_{\dot{j} ^{3} ) \beta_{2j}= \frac{gk_{j} {\omega_{j} (\frac{3+T_{j}^{4} {4T_{j}^{3} ) \gamma_{2j}=k_{j} (\frac{1+T_{j}^{2} {T_{j} )(3.53)For standing waves, since it is not possible to write a_{2j} and b_{2j} in terms of a_{j} or b_{j} , the systemcannot be reduced to a single equation for a_{j} or b_{j} . Therefore, in general, it is necessary tosolve the system given by (3.49)-(3.50) along with (3.27)-(3.28) , except for the infinitely deep
water case, for which a_{2_{j} =k_{\dot{j} a_{j}^{2} as \alpha_{2j}=k_{j} and \gamma_{2j}=2k_{j}.
For a time‐periodic solution, we write a_{j} and b_{j} asa_{j}=A_{j}e^{i\Omega_{j}t}+A_{3j}e^{3i\Omega_{j}t}+C.C. +O(\epsilon^{5}),\Omega_{j}=\omega_{j}[1+\delta_{j}+O(\epsilon^{4})]+O(\epsilon^{5})b_{j}=B_{j}e^{i\Omega_{j}t}+B_{3_{\dot{J}}}e^{3i\Omega_{j}t}+C.C. + O(\epsilon^{5})
where A_{j}=O(\epsilon), B_{j}=O(\epsilon),,,

A_{3j}=O(\epsilon^{3}), B_{3j}=O(\epsilon^{3}) , and \delta_{j}=O(\epsilon^{2}) have been assumed

real. By substituting (3.54)-(3.55) into (3.49)-(3.50) with (3.52), the nonlinear correction to

the wave frequency can be determined at O(\epsilon^{3}) as\delta_{j}=(\frac{9-12T_{j}^{2}-3T_{j}^{4}-2T_{j}^{6} {4T_{j}^{4} )k_{\dot{j} ^{2}A_{\dot{j} ^{2},

(3.56)

which has been obtained by Tadjbakhsh & Keller (1960). As pointed out by Tadjbakhsh &Keller (1960), \delta_{j} is negative for k_{j}h>1.058 , implying that the frequency decreases as the wave
amplitude increases, which is observed for a soft spring. On the other hand, for k_{\dot{j}}h<1.058,\delta_{j} is positive, which corresponds to the case of a hard spring.
For infinitely deep water (harrow\infty, T_{j}arrow 1), (3.56) can be reduced to \delta_{j}=-2k_{\dot{j} ^{2}A_{\dot{j} ^{2} , whichis the result obtained by Rayleigh (1915), where the wave amplitude is defined as \overline{a}_{j}=4A_{j} sothat\delta_{j}=-k_{j}^{2}\overline{a}_{\dot{j} ^{2}/8.

4Conclusion

Using the equivalence between the spectral formulation of Zakharov (196S) and the pseudo‐
spectral formulation of West et al. (1968), we obtain an explicit fifth‐order spectral model thatgoverns the evolution of the Fourier transforms of the surface elevation \zeta and the free surface
velocity potential \Phi . Compared with a lower‐order one, the fifth‐order model would improve

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the description of the spectral evolution of broadband nonlinear waves of finite amplitudes.
When discretized, the model provides a dynamical system for any number of discrete modes,

which would be useful to study nonlinear standing waves in a sloshing tank. Although only thethird‐order solutions for traveling and standing waves have been presented, the fifth‐order time‐
periodic solutions can be easily obtained from the model presented here. It should be remarked
that, as the higher‐order Hamiltonians are also available from the pseudo‐spectral formulationof West et al. (1987) via recursion formulas, it is straightforward to find a higher‐order spectral
model although it would be complicated.

Acknowledgements
The author gratefully acknowledges support from the US National Science Foundation throughGrant No. DMS‐1517456 and OCE‐1634939.

References[1] Annenkov, S. Y. and Shrira, V. I. 2001 Numerical modelling of water‐wave evolution basedon the Zakharov equation. J. Fluid Mech. 449, 341‐371.

[2] Bateman, W. J. D., Swan, C. and Taylor, P. H. 2001 On the efficient numerical simulationof directionally spread surface water waves. J. Comput. Phys. 174, 277‐305.

[3] Choi, W. 1995 Nonlinear evolution equations for two‐dimensional waves in a fluid of finite
depth. J. Fluid Mech. 295, 381‐394.

[4] Choi, W., Kent, C. P. and Schillinger, C. J. 2005 Numerical modeling of nonlinear surface
waves and its validation, Advances in Engineering Mechanics ‐ Reflections and outlooks
in honor of Theodore Y.‐T. Wu, ed. by A. T. Chwang, M. H. Teng & D. T. Valentine,
94‐110, World Scientific.

[5] Craig, W. and Sulem, C. 1993 Numerical simulation of gravity waves. J. Comput. Phys.
108, 73‐83.

[6] Dommermuth, D. G. and Yue, D. K. P. 1987 A high‐order spectral method for the studyof nonlinear gravity waves. J. Fluid Mech. 184, 267‐288.

[7] Clamond, D. and Grue, J. 2001 A fast method for fully nonlinear water‐wave computations.
J. Fluid Mech. 447, 337‐355.

[S] Goullet, A. and Choi, W. 2011 Nonlinear evolution of irregular surface waves: comparisonof numerical solutions with laboratory experiments for long crested waves. Phys. Fluids
23, 016601.

[9] Krasitskii, V. P. 1994 On reduced equations in the Hamiltonian theory of weakly nonlinear
surface waves. J. Fluid Mech. 272, 1‐20.

60
RIMS Workshop on Nonlinear Water Waves[10] Mei, C. C., Stiassnie, M. and Yue, D. K.‐P. 2005 Theory and applications of ocean surface
waves. Part 2: Nonlinear Aspects. World Scientific.

[11] Lord Rayleigh, J. W. S. 1915 Deep water waves, progressive or stationary, to the thirdorder of approximation. Proc. Roy. Soc. Lond. A 91, 345‐353.

[12] Stiassnie, M. & Shemer, L. 1984. On modifications of the Zakharov equation for surface
gravity waves. J. Fluid Mech. 143, 47‐67.

[13] Stokes, G. G. 1847 On the theory of oscillating waves. Trans. Cambridge Philos. Soc. 8,
441‐455.

[14] Tadjbakhsh, I. and Keller, J. B. 1960 Standing surface waves of finite amplitude. J. Fluid
Mech. 8, 442‐451.

[15] Tanaka, M. 2001a A method of studying nonlinear random eld of surface gravity waves bydirect numerical simulation. Fluid Dyn. Res. 28, 41‐60.

[16] Tanaka, M. 2001b Verification of Hasselmann’s energy transfer among surface gravity waves
by direct numerical simulations of primitive equations. J. Fluid Mech. 444, 199‐221.

[17] West, B. J., Brueckner, K. A., Janda, R. S., Milder, D. M. and Milton, R. L. 1987 A New
Numerical method for Surface Hydrodynamics. J. Geophys. Res. 92, 11,803−11,824.

[1S] Whitham, G. B. 1974 Linear and nonlinear waves. Wiley.
[19] Yuen, H. C. and Lake, B. M. 1982 Nonlinear dynamics of deep‐water gravity waves. Adv.
Appl. Mech. 22, 67‐229.

[20] Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of deep
fluid. J. Appl. Mech. Tech. Phys. 9, 190‐194.