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An integrable evolution equation for surface waves in deep water
View the table of contents for this issue, or go to the journal homepage for more 2014 J. Phys. A: Math. Theor. 47 025208
(http://iopscience.iop.org/1751-8121/47/2/025208)
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J. Phys. A: Math. Theor.47(2014) 025208 (17pp) doi:10.1088/1751-8113/47/2/025208
An integrable evolution equation for surface waves in deep water
R A Kraenkel1, H Leblond2 and M A Manna3,4
1Instituto de F´ısica T´eorica (UNESP) Universidade Estadual Paulista (UNESP), Rua Dr Bento Teobaldo Ferraz 271 Bloco II, 01140-070, S˜ao Paulo, Brazil
2LUNAM Universit´e, Universit´e d’Angers, Laboratoire de Photonique d’Angers, EA 4464, 2 Boulevard Lavoisier, F-49045 Angers Cedex 1, France
3Universit´e Montpellier 2, Laboratoire Charles Coulomb CNR-UMR 5221, F-34095, Montpellier, France
E-mail:Miguel.Manna@univ-montp2.fr
Received 22 March 2013, revised 20 November 2013 Accepted for publication 22 November 2013 Published 18 December 2013
Abstract
In order to describe the dynamics of monochromatic surface waves in deep water, we derive a nonlinear and dispersive system of equations for the free surface elevation and the free surface velocity from the Euler equations in infinite depth. From it, and using a multiscale perturbative method, an asymptotic model for small wave steepness ratio is derived. The model is shown to be completely integrable. The Lax pair, the first conserved quantities as well as the symmetries are exhibited. Theoretical and numerical studies reveal that it supports periodic progressive Stokes waves which peak and break in finite time. Comparison between the limiting wave solution of the asymptotic model and classical results is performed.
Keywords: integrable systems, multi-scale methods, deep water, gravity waves PACS numbers: 02.30.Mv, 02.30.Ik, 47.10.−g, 47.35.Bb
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1. Introduction
Wave propagation in an ideal incompressible fluid is a classical matter of investigation in mathematical physics. Especially, surface gravity waves have been intensively studied and many model equations were introduced to handle this problem. Such studies are in a large part motivated by the fact that the initial three-dimensional water wave problem is not tractable
4 Author to whom any correspondence should be addressed.
analytically. Traditionally, the subject was studied in two almost separated domains:shallow water modelsanddeep water models.
The shallow water theory is widely known. The irrotational or rotational Euler equations have been approximated from several sides. An approach is based on the use of two parameters αandβ.α=a/hmeasures the amplitudea of the perturbation with respect to the depthh andβ = h2/λ2 measures the depth with respect to the wavelength λ. Assumingα andβ smaller than 1, a perturbative procedure is carried out. Then model equations are obtained by retaining only the lowest order terms inαandβ[1]. Another well-established approach for the derivation of shallow water model equations is the reductive perturbation method [2–5] based on the Gardner–Morikawa transformation [6]. It enables us to introduce slow space and time variables able to describe the effect of nonlinearity and dispersion asymptotically in space and time. An alternative method was introduced by Serre [7] and several years later by Su and Gardner [6] and Green and Naghdi [8–10]. It is based on an Ansatz which uses the shallow water limit of the exact linear solution of the Euler equations. Henceforth we will name this Ansatzthe linear pattern Ansatz.
This way were obtained the nonlinear shallow water [1], the Boussinesq [11, 12], the Korteweg–de Vries (KdV), the modified KdV (mKdV) [13,14], the Kadomtsev–Petviashvili [15], the Benjamin–Bona–Mahony–Peregrine [16,17], the Serre, the Green–Naghdi and more recently the Camassa–Holm [18] equations. All these model equations govern the asymptotic dynamics ofwave profiles of long waves in shallow water.
The deep water limit consists in considering an infinite depth (h = −∞) in the Euler equations. Then neitherαnorβcan be defined. The Gardner–Morikawa transformation, based on the dispersionless shallow water linear limit of the Euler equations, cannot be defined any more because the deep water limit is fully dispersive. Consequently, surface wave propagation in deep water is mainly concerned with the nonlinear modulation of wave trains. The most representative model equation is the ubiquitous nonlinear Schr¨odinger equation (NLS) [14,19,20]. For a full account on modulation of short wave trains in water of intermediate or great depth see [21]. However, some model equations have been given for wave profile evolution in deep water. E.g., in [22] is derived a finite-depth Boussinesq-type equation for the profile of irrotational free surface waves. The derivation is carried out via the theory of analytic functions followed by a perturbation procedure. Its deep water limit is exhibited as a function of integral operators. The Benjamin–Ono equation, first introduced by Benjamin [23] and Ono [24], is worth being mentioned. It is a nonlinear partial integro-differential equation which describes one-dimensional internal wave profile evolution in deep water. In [25] an approach analogous the one we develop in the present paper was introduced. Other model equations whose solutions behave as deep water waves can be found in [26] and [27]. Their dispersion relations coincide exactly with that of water waves in infinitely deep water and the nonlinear terms are chosen in aad hocway in order to reproduce the Stokes limiting wave.
The present work is the extension of a long sequel of studies carried out by the authors in various physical contexts, dealing with model equations for nonlinear and dispersive short- wave dynamics ([28–42]). Especially, the purpose of the paper is to study the dynamics of an elementary wave profile in deep water instead of looking for modulation dynamics of the wave train. To reach this goal, we seek for the combined effects of dispersion and nonlinearity on a givenFourier component with wave vectork. Our approach is a generalization of the linear pattern Ansatz first used for surface waves in shallow water. We assume an Ansatz based on the exact linear solution of the Euler system for deep water. Of course, as in the shallow water case, the Ansatz does not yield an exact solution of the full Euler system. It produces a depth-averaged model equation, which is expected to be valid close to the linear limit of the Euler system.
The model is able to reproduce, with a good degree of accuracy, the main quantitative results concerning the Stokes wave: limit phase velocity, limit aspect ratio, Stokes’ greatest height and angle. It is completely integrable with an associated Lax pair and in this sense it is the analogue for short wave in deep water of Korteweg-de Vries for long wave in shallow water.
In the NLS approximation, the underlying periodic wave remains close to sinusoidal.
Although the first harmonics are taken into account in the derivation of the NLS, they are assumed to remain small, and higher harmonics are neglected. In shallow water, highly non- sinusoidal periodic waves are well known: these are the cnoidal wave solutions to KdV. We present here the deep water counterpart of such waves.
The paper is organized as follows. In section (2) we introduce the Euler equations in deep water. After a brief summary of the ‘modus operandi’ of the linear pattern Ansatz in the shallow water limit, we generalize it to the deep water case. In section (3) we give the nondimensionalization of the Euler equations and we introduce the deep water surface wave problem. In section (4) we use the linear pattern Ansatz in the deep water context and we derive a nonlinear and dispersive system of equations for the free surface elevation and for the free surface velocity. In section (5) we derive a small wave steepness surface wave model equation in deep water. In section (6) are exhibited the mathematical properties of the model:
the Lax pair, the relation with the integrable Bullough–Dodd model and the symmetries. In section (7) is studied, both theoretically and numerically, the progressive periodic Stokes wave.
In theappendixwe give some detail on the derivation of the Bullough–Dodd equation and finally section (8) draws the conclusions.
2. A deep water linear pattern Ansatz
We considered the dimensional Euler equations for deep water. Let the particles of the fluid be located relative to a fixed rectangular Cartesian frame with originO and axes (x,y,z), where Ozis the upward vertical direction. We assume translational symmetry alongyand we will only consider a sheet of fluid parallel to the xz plane. The velocity of the fluid is V(x,z,t)=(U(x,z,t),W(x,z,t)). The fluid sheet is moving on a bottom atz= −∞, and its upper free surface is located atz=η(x,t). The continuity equation and the Newton equations (in the flow domain) read as
Ux+Wz=0, ρU˙+Px∗ =0, ρW˙ +Pz∗+gρ=0, (1) in which subscripts denote partial derivatives and the dot (˙) denotes the material derivative defined byF˙ =Ft+U Fx+W Fz. The boundary conditions atz= −∞and atz=η(x,t)are W =0 forz→ −∞and
P∗−P0=0 and ηt−W +Uηx=0, both for z=η, (2) whereρis the constant density of the water,gthe gravitation constant andP0the atmospheric pressure. The solution of the problem consists in findingη(x,t),V(x,z,t)andP∗(x,z,t).
Approximate solutions to the nonlinear surface water waves problem can be computed only if thez-dependence of the velocity fieldV(x,z,t)is known. The z-dependence provides a coupling between the wave motion at the surface and the wave motion at various depths. In the linear case thez-dependence is known, hence the solution is exactly known. Expansions in power series of zas well as the theory of harmonic functions have been used to solve this problem in the nonlinear Euler equations. Another approach to the problem is the use of an Ansatz based on the linearz-dependence ofV(x,z,t). It isthe linear pattern Ansatz introduced in [6–9], in the shallow water context, and equivalent to the widely knowncolumnar
hypothesis. It is thus worthy to begin with a brief description of how the Ansatz works in the shallow water case, before we extend it to the deep water context.
2.1. The shallow water Ansatz
For surface progressive waves of wave vectorkin water of depthhin the shallow limitkz∼0, the linear solutionUlto the horizontal componentUof the velocity and the dispersion relation ωare
Ul =Aexp i(kx−ωt)[1+O(kz)2+ · · ·], (3)
ω=k gh
1−16(kh)2+ · · ·
, (4)
whereAis a constant, in complex representation so that the physical velocity is given by real part of equation (3).Ulis a wave crest moving at the phase velocityc(k)given by
c= gh
1−16(kh)2+ · · ·
. (5)
The linear pattern Ansatz assumes that thez-dependence ofUis the same as thez-dependence ofUl, and that its spatiotemporal behaviour is given by an undetermined functionu(x,t). So, it assumes thatU =u(x,t)which corresponds to the extreme long wave limit in (3), i.e.,U does not depend onz. Since the phase velocity is invariant underk→ −k, i.e.,c(k)=c(−k), the Ansatz does not need to take into account the direction of propagation inu(x,t).
2.2. The deep water Ansatz
We propose in the present paper an analogous procedure in the deep water case. Nevertheless two important differences arise. The first one is that in the deep water case we have
Ul =Aexp i(kx−t)expkz, =
gk, c= g
k. (6)
So, the phase velocitycis no longer invariant underk→ −kand expressions (6) are strictly valid for wave crests moving to the right only. Consequently, the Ansatz would take into account the choice of the propagation direction made in deriving the linear solution. The second difference is a very subtle one. Here we are interested in the effect of nonlinearity and dispersion on a purely sinusoidal wave of wave vectorkand frequencyfor largexandt.
However, wave vectors and frequencies are no longer constant in nonlinear system, and are generalized to the concepts of local wave vectork(x,t)and local frequency(x,t). There will then be a non-uniform local wave train (nearly sinusoidal) withk=k(x,t)and=(x,t). It can be shown [1,43] thatk(x,t)and(x,t)remain constant for an observer moving at the group velocitycg(k). Thus, to study the asymptotic behaviour for largexandtof a wave with given wave vectorkin deep water, the analysis must be carried out in the frameR(cg) travelling at the velocity
cg=1 2c= 1
2 g
k, (7)
with respect to the frameRwhere the Euler equations were written. Therefore, the linear pattern Ansatz we propose is
U=U(x,z,t)=u
k
x−1 2ct
,t
expkz, c=c(k)= g
k. (8)
This Ansatz assumes that all Fourier components, i.e. all harmonics of the wave, since we consider a periodic wave, have the same extension inz, which is strictly speaking not true.
Consider first some higher harmonic, say thenth one. In fact it can be of two kinds: thenth harmonics generated by the fundamental wave at each point in space and time, and thenth harmonic resulting from the propagation of the former. (This has been proved in the case of second harmonic generation, first from multiscale expansions [44], and more recently it has been observed experimentally [45].) The generatednth harmonic is proportional toUn, hence itsz-dependence is of the form expnkz, corresponding to a depth 1/nk. The propagatingnth harmonic has a wave vectornk, and consequently the samez-dependence and depth. This matching is essential in the influence of the harmonics on the wave, it is kept by the averaging, since only the powers of the profileuwill compare to the harmonics. For a small deviation of the wave vectork, however, the deep water Ansatz will not take into account the variation of the wave characteristics due to the change in depth. Hence corrections would have to be taken into account to get correct account of phenomena such as modulational instability from this Ansatz.
3. The deep water surface wave problem
The first step is put the Euler equations in dimensionless form. Hence the original variables of space, time, velocity components and pressure will be nondimensionalized withk,√
√ kg,
k/gandk/ρgrespectively.kis the wave vector of the wave to be described. Hence the Euler equations for−∞<z< ηbecome
Ux+Wz=0, U˙ = −Px∗, W˙ = −Pz∗−1, (9) and the same expressions as above for the boundary conditions. We integrate the Euler equation in the depth of the fluid. From the continuity equation and its boundary condition we get:
W = −z
−∞Ux(x, ζ ,t)dζ. This expression forW allows us to write the material derivatives U˙ andW˙ and to findP∗. With these and using the Leibnitz’s rule inU˙ = −Px∗we obtain Ut+UUx−Uz
z
−∞Uxdζ= −ηx+ η
z
dζ ζ
−∞Uxxtdζ+ηx η
−∞Uxtdζ + η
z
Udζ ζ
−∞Uxxxdζ+ηxU(η) η
−∞Uxxdζ
− η
z
Uxxdζ ζ
−∞Uxdζ−ηxUx(η) η
−∞Uxdζ , (10) withU(η)=U(x,z=η,t). Finally the equation forηtexpresses as
ηt+U(η)ηx+ η
−∞Uxdζ =0. (11)
If thez-dependence ofU(x,z,t)was known, the surface wave problem in deep water could be solved. Equations (10) and (11) yield a system of coupled nonlinear equations forU(x, η,t) andη(x,t), which is the complete solution of the problem. However this is not the case because U(x,z,t)is unknown versusz.
4. Az-averaged system of equation
Thez-dependence ofU(x,z,t)is the central issue making unsolvable the nonlinear surface water wave problem in deep as in shallow water. Consequently we use the deep water Ansatz (8), which reads in dimensionless variables as
U(ξ,z,t)=u(ξ,t)expz, with ξ =x−12t. (12)
It implies the change in the differential operators: ∂/∂t → −(1/2)∂/∂ξ +∂/∂t, and
∂/∂x→∂/∂ξ. Then we obtain from (10) the expression
utexpz−12uξexpz+ηξ = −uξξtexpz+(uξtexpη)ξ +12u(uξξexp 2η)ξ
−12uuξξξexp 2z−12uξ(uξexp 2η)ξ
+12uξuξξexp 2z−12(uξξexpη)ξ +12uξξξexpz (13) This equation is now taken into account through an averaging over the full depth of the fluid, defined by the expression
z0→−∞lim 1 η−z0
η z0
{∗}dζ . (14)
Applying (14) to (13) we obtain uξt−12uξξ
exp(η)
ξ+12
exp(2η)
uuξξ−u2ξ
ξ−ηξ =0. (15)
Finally, substituting (12) into (11), we get
ηt−12ηξ +(uexpη)ξ =0. (16)
Note that we did not assume that eitherηoruvanish asξ → ∞. The system (15) and (16) forη(ξ,t)andu(ξ,t)is az-average of the Euler equations resulting from the flow hypothesis (12) and the subsequent average in the depth (14). It isa deep water analogousof the Serre or Green–Naghdi or Su–Gardner system. It differs from the result obtained in [25]. The latter is aweightedz-integration of the Euler equations. The crucial step there was the use of a precise weight in order to regularize the Archimedean divergence term present in the pressure. In turn this produces aweight-dependent modelthe degeneration of which is eliminated by the requirement that the linear limit possesses phase and group velocities equal to those of the deep water system. The present derivation does not need either the weighted average procedure or thead doc hypothesisof equality between the linear limits. Both of them,z-average and weightedz-integration, coincide in the classicalshallow watercase.
5. A small wave steepness surface wave model
Up to this point the asymptotic behaviour inξ andt of the surface wave profile was not considered. This can be performed by taking advantage of some small parameter present in the system. Therefore, let us rewrite the dimensionless functionη asη(ξ,t) = kaH(ξ,t), where the dimensionless functionH(ξ,t)represents the surface deformation with amplitude ameasured from the initial mean water levelz=0, and the productka =is thesteepness coefficient(oraspect ratio)which measures how much the wave profile is peaked ( >1) or flat ( <1). The equations (15) and (16) are written explicitly in terms ofandHas
uξt−12uξξ
exp(H)
ξ +1
2exp(2H)
uuξξ−u2ξ
ξ −Hξ =0, (17)
Ht−12Hξ +[uexp(H)]ξ =0. (18)
In fact, (17) and (18) are valid for any order in , but they are dispersive and strongly nonlinear, so untractable analytically. Nevertheless, a perturbative theory can be carried out in terms ofby considering flat propagating waves, i.e., <1. The linearized system yields the dimensionless form of the dispersion relation=1/2. Nonlinearity causes deviations from it (Stokes’ hypothesis), which can be taken into account through=1/2++2+· · ·.Hence the phase becomesξ−t=ξ−1/2t−t−2t+ · · ·,and we can define new variablesy,τ, ν,. . ., as:y=ξ−1/2t, τ =t, ν=2t, . . . .The functionH(ξ,t)must then be regarded
as a function of the new independent variablesy,τ,ν,. . .[46,47], i.e., asH(y, τ, ν, . . . , ).
On the other hand, the derivation operators with respect toxandtbecome∂/∂ξ =∂/∂y,and
∂/∂t= −(1/2)∂/∂y+∂/∂τ+2∂/∂ν, . . .. Since exp(H)=1+H+O(2), we can retain in equations (17) and (18), after the leading order0,bothorders0 and, and so on. Thus, equation (18) becomes
−12∂y+∂τ+O(2)
H−12∂yH+[u(1+H+O(2))]y=0. (19) This equation givesuin terms of derivatives and antiderivatives ofH, as
u=H−2
H2+
y
−∞Hτdy
+O(3), (20)
where it was not assumed that eitheruorH(and derivatives) vanish asy→ −∞. Reporting the expression ofuinto (17) and retaining terms inand2, we obtain the equation
−22Hτyy+Hy+Hyyy−322(HHy)yy =0, (21) which in terms of the dimensionless variablesξ,tandη(ξ,t)reads as
2ηξξt=ηξ −32(ηηξ)ξξ, (22)
or in terms of variables with dimension:
2
k
gηξξt=k2ηξ−3
2k(ηηξ)ξξ. (23)
This equation is the main result of our work. It describes the asymptotic nonlinear and dispersive evolution of small steepness waves of a Fourier wave vectorkin deep water. It has k-dependent coefficients. Consequently we have aninfinite set of equations indexed by the wave numberk. Equation (23) can be considered as belonging to both of the two categories:
that of KdV models (KdV, mKdV, Benjamin–Bona–Mahony–Peregrine, Camassa–Holm, etc) describing evolutions of wave profiles and that of NLS-type equations (modified NLS [14], Davey–Stewartson [48], etc) describing modulation of wave profiles and havingk-dependent coefficients.
6. Mathematical properties
6.1. Lax pair
In this section we assume thatηand its derivatives vanish asξ → ±∞. Equation (22) has the dispersion relation=1/2, as required, because (22) was derived in a frame moving at the group velocity associated with the Euler equations in deep water. Making use of the scalings η(ξ,t)−→(2/3)η(ξ,t)and 2∂/∂t −→∂/∂t, then equation (22) becomes
ηξξt =ηξ −(ηηξ)ξξ, (24)
and integrating once yields
ηξt=η−(ηηξ)ξ. (25)
Equation (25) was already derived in other contexts (see [49–51]). In [52] was shown that it can be derived as a short-wave limit from the Degasperis–Procesi equation [53]. Wave breaking was studied in [54].
Let us consider the functionF defined by
F3=1−3ηξξ, (26)
or in terms ofφ=ηξξ, byF3=1−3φ. One of the most remarkable properties ofFis that it allows us to build non-trivial conserved quantities. The two first ones are
Ft= −(ηF)ξ,
2φ2ξF−7
t=
(1+φ)F−1−2ηφξ2F−7
ξ. (27)
The Lax pair also can be constructed by means ofF and reads Lˆ = ∂
∂ξ +iλFσˆ3+1 2
ηξξξ
F3 σˆ1, Mˆ = −1
2 ηηξξξ
F3 σˆ1−iληFσˆ3− i 8λ
1−ηξξ F σˆ3+1
4 ηξξ
λFσˆ2, (28) where σˆ1,σˆ2 and σˆ3 are the Pauli matrices, λ the spectral parameter and equation (25) is obtained as the classical Lax equationLˆt=[L,ˆ M].ˆ
6.2. Reduction to the Bullough–Dodd equation
Through the change of variables from ξ to pand the change of functions from η(ξ,t)to r(p,t)defined byp =ξ
Fdξ and 1−3ηξξ = exp(r), it is found thatr(p,t)satisfies the Bullough–Dodd equation (see theappendix)
rpt= 19 exp2
3r
−exp
−13r
. (29)
The Bullough–Dodd equation was introduced in [55,56], in which some non-trivial conserved densities were also exhibited. Its Lax pair and complete integrability were shown in [56]. The change of variable and of function leading from (25) to (29) was introduced in the context of a systematic study of short-wave dynamics in long-wave model equations, in [30,33,38,57].
Equation (24) is invariant under the discrete transformationη(ξ,t)→η(−ξ,−t), and can be written as the conservation law
(ηξξ)t =(η−(ηηξ)ξ)ξ. (30)
It is invariant under the Galilean group of transformations
t=t, x=ξ +V t, η(ξ,t)=η0+s(ξ,t), (31) in whichV andη0are constants withη0= −V. In this case indeed, equation (24) transforms intosxxt =(s−(ssx)x)x. Consequently, (24) must be seen as a member of a set of equations parametrized by the speedV of the Galilean group of transformations (31). The integrated equation (25) has the Lorentz invarianceξ →κξ,t →t/κ, η→κ2η,withκ a real arbitrary parameter.
Equation (25) belongs to the family of equations Sxt = −S−SSxx+K+1
2 (Sx)2, (32)
K being a real constant. Equation (32) describe the asymptotic dynamics of short waves.
WithS = −η,x → −xandK = −3, equation (32) reduce to our main equation (25). The equation of the family corresponding toK=2 is the short-wave limit of the Camassa–Holm equation associated with geodesic flow onN-dimensional quadrics [58]. ForK=1 it governs short surface wind-waves [32] and has peakons and stationary compacton solutions. In both cases the equation of the family (32) is associated with equations belonging to the very wide Bullough–Dodd family (the caseK=2 is associated with the sine-Gordon equation and the caseK=1 with the Bullough–Dodd–Zhiber–Shabat one).
The reductions of equation (32) provide evolution equations with two main behaviours:
they are connected with members of the Bullough–Dodd family and possess solutions continuous inxbut having finite discontinuity in the firstx-derivatives as peakons, compactons and Stokes limiting wave. For peakons thex-derivative discontinuity is propagated as in the present case for the Stokes limiting wave, see below.
7. Periodic wave and limiting wave
7.1. Harmonic solution and perturbative approach
A periodic wave solution of (23) is predicted by linear analysis. It is sinusoidal, of small amplitude and travels with phase velocity(1/2)√
g/k. The question is: up to what values of its amplitude and velocity does the periodic wave exist? This is still possible for wave amplitudes large enough so that higher terms in the perturbation series can no longer be neglected in equation (23). Then the wave is not exactly sinusoidal, its velocity is not exactly 12√
g/k, and thelimiting wavehas the greatest height before breaking. In order to find thissteep rotational Stokes wave, let us start with equation (23) in the frameRwith coordinates(x,t), which reads as
2
k
gηxxt=k2ηx−ηxxx−9
2kηxηxx−3
2kηηxxx. (33)
The radius of curvature is defined byR=[1+(ηx)2]3/2/ηxx, hence equation (33), within the allowed order, and assuming a progressive periodic wave in the variablez=x−ct, reduces to
2c
k
gRz=k2ηzR2+Rz−9
2kηzR+3
2kηRz. (34)
Letηbe the height of the wave, i.e.,η=ηcrest−ηtroughand assume that the wave approaches the limiting wave of heightη=ηmaxand that this value is reached forz=z0; i.e.,η(z0)=ηmax
withz0=x0−cLt0, wherecLis the phase velocity at the limit. At the limit, on the one hand, the radius of curvatureR(z0)is zero [59,60], and on the other hand its derivativeRz(z0)is not well defined. Nevertheless, equation (34) can be satisfied if
2cL
k
g=1+3
2kηmax. (35)
To determine the limitkηmaxandcL, one further equation is required. This equation will come from the Stokes series for the periodic solution of small amplitude of equation (33). Hence we consider a periodic solution η(z) = δ
η0(z)+δη1(z)+δ2η2(z)+ · · ·
withδ a small amplitude defined asδ= 12(ηcrest−ηtrough)= 12η.To avoid secular terms, the velocitycmust be also expanded asc=c0+δc1+δ2c2+ · · ·. We write equation (33) in terms ofz, report expansions of ηand cinto it, and eliminating order by order secular terms we obtain the solutionη(z)at order as
η(z) =δ
cos(kz)+(δk)
2 cos(2kz)+27(δk)2
32 cos(3kz)+ · · ·
, with c=
g k
1+3(δk)2 16 + · · ·
. (36)
Recall that Stokes found, for the same quantities and using the present notations,the expansions η=δ
cos(kz)+(δk)
2 cos(2kz)+3(δk)2
8 cos(3kz)+ · · ·
, c=
g k
1+(δk)2 2 + · · ·
. (37)
It is seen that both solutions coincide up to first order in the small quantity = kδ, while differences are seen from order2on. A part of the discrepancy may come from the fact that Stokes’ results were obtained for irrotational waves, while we did not require that the fluid is
irrotational. Further, we use an averaging of the Euler equation over the depth, in such a way that the effects of small variations of the wavelength on the effective depth of the wave are neglected. As mentioned above, this has consequences on the accuracy with which dispersion is accounted for by our model, which may result in a part of the observed discrepancy. Now, when the solution (36) approaches the limiting wave solution we must haveδ −→ηmax/2, hence
cL
k
g =1+3(kηmax)2
64 . (38)
This is the second equation we were looking for. From (35) and (38) we obtain kηmax=0.697, cL
k
g =1.022. (39)
The classical result found a long time ago by Michell [61] forkηmaxand recent computations forcL[62] yield the values
kηmax=0.892, cL
k
g =1.092. (40)
Comparing (39) and (40) we find the percentage errors (100% times the relative errors) (kηmax)error= −21%,
cL
k g
error
= −6,4%. (41) Deviations of kηmax andcL, calculated from (33) in relation of (40) are due to two mains factors:(1)the results (39) come from a computation at second order inδ only, while (40) were obtained using theories at least at fifth order inδ,(2)the classical results were obtained from the complete Euler equations,
7.2. Exact analytical approach to the limiting wave
We start with the equation (22) in the referentialRrelative to the variables (ξ,t), i.e.,
2ηξt =η−ηξξ−34(η2)ξξ. (42)
We look for a travelling wave with speedc:η=η(z)withz=k(ξ−ct). Equation (42) is then multiplied by
(3/4)η2+aη
zand integrated once, which yields 3
4η2+aη
z
2
=η3+aη2+K, (43)
where we have set a = 1−2c, and K is some integration constant. Using separation of variables and settingη=aY,z=Z√
|a|andK= |a|3ptransforms (43) into
dZ= ± 3Y +2
2
p+ε(Y3+Y2)dY, (44)
εbeing the sign ofa. Then we look for some regular periodic solution. It attains its minimal valueY1for someZ, sayZ =Z1, and then will grow up to some maximal valueY2, reached atZ =Z2for the first time afterZ1. dY/dZmust be well-defined, real and positive for allZ betweenZ1andZ2and allY betweenY1 andY2. Further dY/dZmust be zero forZ =Z1and Z2. The conditionsZ1<Z2andY1 <Y2must also be satisfied. f(Y)=Y3+Y2presents two local extrema, f(−2/3)=4/27 and f(0)=0. Hencep+εf(Y)has more than one zero ifp is between 0 and−ε4/27. Since dZ/dY is zero forY = −2/3, and dY/dZmust remain finite over the whole interval, the valueY = −2/3 is excluded, andY1andY2must be the two largest
Figure 1.The normalized wavelengthZ2−Z1of the Stokes wave versus its normalized amplitudeY2−Y1.
zeros ofp+εf(Y), withY1 <0 <Y2. Forε= +1, p+εf(Y)is negative in this interval.
Hence, since its square root must be defined, the caseε =1 is excluded. The period of this solution, which is the wavelength of the Stokes wave in normalized units, is
Z2−Z1 =
Y2 Y1
3Y+2 2
p−Y2−Y3dY. (45)
The amplitudeY2−Y1is given by the solution of the cubic polynomial equationY3+Y2−p=0, which can be computed explicitly from algebraic solution. The integral (45) can be computed numerically using standard methods. Finally the wavelength is shown versus the amplitude in figure1. The maximum amplitude it attained with the minimal wavelength for p=4/27.
Then the quantityp−Y2−Y3factorizes explicitly, and after simplification we get dZ
dY = 3√ 3 2
√ 1
1−3Y, (46)
which is straightforwardly integrated, and then inverted, to yield Y = 1
3−Z2
9 , (47)
in which we have set arbitrarilyZ1 = 3 (translation invariance). Formula (47) is completed using parity and periodicity to
Y = 1 3−1
9
Z−6E Z+3
6 2
, (48)
Edenoting the integer part. Coming back to the original variables, we obtain the expression of the limiting Stokes wave, as
η= λ2 36
2z λ −2E
z λ +1
2 2
−1 3
, (49)
in whichλ=6√
−ais the wavelength (it has been seen thatε=sign(a)= −1). The velocity isc=1/2−a/2=1/2+λ2/72.The total amplitude is−a=λ2/36.
Coming back to the variables with dimension, we obtain η=kλ2
36 2ζ
λ −2E ζ
λ+1 2
2
−1 3
, (50)
Figure 2.The limiting Stokes waves in dimensional form, forλ=1 m.
in whichλ =λ/kis the wavelength with dimension andζ =x−(k3/2λ2√
g/72)t. We need to identifyk. The exact linear wave must be periodic with wavelengthλl =2π/k. Assuming the same value ofkfor the same wavelength in the nonlinear case (λ=λl), we get (dropping the prime)
η=2πλ 36
2ζ λ −2E
ζ λ+1
2 2
−1 3
, (51)
andζ =x−cLt, with cL
k g =
1 2+π2
18
1.0483. (52)
The limiting Stokes waves, in its dimensional form (51), is plotted on figure2forλ=1 m (recall thatY/η <0.) Now the percentage errors using (51) and (52) are(kηmax)error = +22 and(cL
k
g)error= −4%.
7.3. The evolution of the limiting Stokes wave
The evolution of the limiting Stokes wave is checked numerically. We use the equation (42), written in frame traveling at velocity 1, as
2ηξt =η−34(η2)ξξ. (53)
Equation (53) is integrated with respect tox, then we apply the Fourier transform defined as
y(ξ,t)= y(σ,ˆ t)e2iπσ ξdξ, (54)
to get
ˆ ηt = 1
4iπσηˆ−3
4iπση2. (55)
Then the time evolution of ηˆ is computed by means of a standard fourth order Runge–
Kutta algorithm. The Fourier transforms are computed using a standard fast Fourier transform algorithm. It must be noticed that the term representing the antiderivative in equation (55) is not defined forσ =0. The corresponding term in the discrete scheme is set to zero, which assumes a zero mean value. The mean value of the limiting Stokes wave (51) is easily computed, it
0 20
40 60
80 100
-1 -2 1 0
2
ξ
-0.01 0 0.01 0.02
η
Figure 3.The evolution of the limiting Stokes wave, as computed numerically. The initial data att=0 is given by equation (49) with the dimensionless wavelengthλ=1.
0 2
4 6
8
-8 -10 -4 -6
0 -2 4 2
ξ -1
0 1 η
Figure 4.Evolution of the free surfaceηaccording to equation (53), starting from the sinusoidal input of equation (56) with amplitudeA=1 and angular frequencyω=1.
is zero as required. Further, the fact that (51) is not continuously derivable induces a high- frequency numerical instability. The latter is removed using a spectral filter in the numerical scheme, which attenuates the highest frequencies. It is checked that this filter does not affect the spectrum of the limiting Stokes wave itself. The result of the computation is plotted on figure3.
7.4. From sinusoidal wave to the limiting wave
The evolution equation is solved numerically as above, starting from an initial data in the form of a purely sinusoidal wave,
η(ξ,0)=Acosωξ. (56)
If the initial amplitude is large enough, the wave evolves to the limiting wave, see figure4.
The profile obtained for a set of values of the amplitude is shown on figure5. It is seen that the velocity grows with the amplitudeA. ForA=0.1 the profile remains close to a sinusoidal one. It is still smooth forA=0.2, but progressively changes to a sharp edge asAincreases. It must be noticed that the sharp edge of the profile appears in practice well below the amplitude reaches the threshold yielding the limiting wave, in accordance with usual observation. Figure6
Figure 5.The wave profile (free surfaceη) att=100, starting from sinusoidal inputs as in equation (56) with angular frequencyω=1 and amplitudesA=0.1, 0.2, 0.3,. . ., 1. The rightmost crests correspond to the highest values ofA, according to the fact that wave velocity increases with amplitude.
Figure 6.Evolution of the maximum and minimum height of the free surfaceη, starting from sinusoidal inputs with angular frequencyω=1 and amplitudesA=0.1, 0.2, 0.3, . . ., 1. The curves shown here come from the same numerical data as figure5, and the same colour code is used.
presents the evolution in time of the maximum and minimum values of the wave profile, i.e.
of the free surface elevationη, for the same set of values of the initial amplitude as in figure5.
A first fast transient corresponds to the arising of a shock profile, which then evolves more slowly towards the nonlinear Stokes wave, with some additional modulation for moderate amplitudes. The periodic modulation arises from the generation of harmonics at the beginning of the propagation, and their interferences; this feature could be evidenced by a perturbative approach about the sinusoidal wave. Other features could not be observed by perturbation of the sinusoidal profile, especially the saturation of the value of the maxima and minima to the crest and the though of the limiting wave. It is indeed seen that the profile saturates to the limiting wave one as soon as the amplitudeAattains the value 0.6. The relaxation to this state takes some time; the sharp peak, aboutξ =5, on the blue line which corresponds toA=0.6 on figure5, is the trace of the discrepancy between the limiting profile and the real one in the transient regime. It eventually vanishes completely.
8. Conclusions and final comments
We have obtained adeep water equivalent of the widely known Serre or Green–Nagdhi or Su–Gardner shallow water system of equations. This was done via an appropriated Ansatz for the vertical coordinate dependence of the velocities. This procedure allowed us to describe the nonlinear and dispersive dynamics of the wave profile itself, avoiding the classical route of wave modulation of a short wave train. From the system, we were able to derive an asymptotic model equation (in times and space) forshort waves. The new model is the short wave analogous of KdV in surface waves and, as KdV, shown complete integrability. The progressive Stokes wave was studied analytically and numerically. The results for the limiting wave height and the limiting phase velocity, within the degree of approximation of the model, are in accordance with classical results. Important future works and open issues to be study are (a) the analysis of the Stokes limiting wave of the system (15) and (16), (b) the many perspectives in the antidifusive systems associated with wind-wave generation, (c) inclusion of surface tension, small viscosity as well as higher spatial dimension, (d) the achievement of asymptotic systems of higher degree than 2 in the aspect-ratio parameter who can answer the issue: is the new asymptotic model the first term of a hierarchy of integrable models, as KdV is?
Acknowledgments
MAM thanks the PVE program (Pesquisador Visitante Especial, CAPES/BRASIL) and ANR- Project MOROC’H (ASTRID 2013).
Appendix. Reduction to the Bullough–Dodd equation
The change of variables fromξtopand the change of functions fromη(ξ,t)tor(p,t)defined in subsections6 and6.2together with the conservation of F,Ft = −(ηF)ξ, gives for any functionM(ξ,t)the relations
∂M(ξ,t)
∂ξ =F∂M(p,t)
∂p , (A.1)
∂M(ξ,t)
∂t = −(ηF)∂M(p,t)
∂p +∂M(p,t)
∂t . (A.2)
From (A.2) follows that
∂2M(ξ,t)
∂t∂ξ +η∂2M(ξ,t)
∂ξ2 = −ηξF∂M(p,t)
∂p +F∂2M(p,t)
∂p∂t . (A.3)
We take the derivative of the evolution equation (25) with respect toξ, once and then twice.
Using the definition ofF, it yields
ηξξt+ηηξξξ =ηξF3, (A.4)
ηξξξt+ηηξξξξ =ηξξF3−4ηξFηξξp. (A.5) Equations (A.1) and (A.2) forM=ηξξ are reported into (A.4), which gives
ηξ = 1
F3ηξξt. (A.6)
Then using (A.3) forM =ηξξ, and (A.5) and (A.6), brings to the important relation
ηξξptF3=F5ηξξ−3ηξξtηξξp. (A.7)
The change of functions and the expressions ofF,FpandFtgive rpt= −1
3
F3ηξξpt+3ηξξtηξξp
F6
. (A.8)
Finally using (A.7) and the expression ofFwe obtain the Bullough–Dodd equation (29).
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