Interaction of modulated gravity water waves of finite depth

0.42 MB·

by  Ioannis Giannoulis

INTERACTION OF MODULATED
GRAVITY WATER WAVES OF FINITE DEPTH
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1
IOANNISGIANNOULIS
0
2
Abstract. We consider the capillary-gravity water-waves problem of finite
n
depth withaflat bottom of one ortwo horizontal dimensions. We derive the
a
modulation equations of leading and next-to-leading order in the hyperbolic
J
scalingforthreeweaklyamplitude-modulatedplane-wavesolutionsofthelin-
1
earized problem in the absence of quadratic and cubic resonances. We fully
3
justifythederivedsystemofmacroscopicequationsinthecaseofpuregravity

waves, i.e. in the case of zero surface tension, employing the stability of the
]
P water-wavesproblemonthetime-scaleO(1/ǫ)obtainedbyAlvarez-Samaniego
andLannes.
A
.
h
t
a 1. Introduction
m
Asignificantpartintheresearchonwaterwavesisbasedonthestudyofasymp-
[
totic limits derived from the original water-waves problem, which is considered as
1
providingthe complete descriptionof their behavior. The benefit of this method is
v
that these reduced models have, on the one hand, a clearer structure and, on the
5
5 other hand, they highlight a particular qualitative feature of the wave evolution.
2 Depending on the aspect of the nature of the water waves one is interested in,
0 one has to employ the relevant asymptotic scaling, obtaining in each case a differ-
0
ent macroscopic limit. Considering that from the outset the original water-waves
.
2 problem has some fundamental characteristics, the result is a plethora of different
0 equations which are presumed to describe approximatively the behavior of water
6
waves in different situations and regarding different aspects. While the choice of
1
the relevant asymptotic scaling requires a thorough understanding of the initial
:
v
model, from an analytical point of view the crucial question is the justification of
i
X the derived model, i.e. the proof that its solutions indeed are approximations of
r solutions to the original problem.
a Concerning the initial set-up of the water waves problem one could distinguish
roughly between (a) gravity or capillary-gravitywaves (the latter ones taking into
account together with gravity also the surface tension as driving forces for the
evolution of the waves), (b) finite- or infinite-depth water, and (c) two- or three-
dimensional space, where in the former case one considers a vertical plane in the
water domain that contains the (dominant) direction of evolution, assuming that
the waves are (nearly) constant in the direction normal to the plane. Of course,
this is only a very rough classification(e.g. in the case of finite depth one can con-
sider shallow or deep water, flat bottoms or bottoms with some (smooth or rough)
topography, or even moving bottoms etc.), but it seems to be the prevalent one in
the mathematical-analytical literature, where the fundamental question concerns
Date:February2,2016.
2010 Mathematics Subject Classification. Primary76B15;76B45, 35B35,35L03,35L05.
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2 IOANNISGIANNOULIS
the well-posednessofthe water-wavesproblemandinparticularthe existence time
of its solutions.
In the case of gravity water waves in two dimensions and for infinite depth,
first local well-posedness results were obtained by Nalimov [41] in 1974 for small
Sobolev initial data, and by Shinbrot [48] and Kano and Nishida [32] for analytic
initial data. The method of Nalimov was employed to prove local well-posedness
in the case of finite depth by Yosihara [56] and by Craig [11], who obtained also
first rigorous justification results of the Korteweg-de Vries (KdV) and Boussinesq
approximations. However, in the case of infinite depth, the crucial breakthrough
was made by the work of S. Wu, who presented localwell-posedness results for the
two-andthree-dimensionalcaseswithoutsmallnessassumptionsontheinitialdata
in[52],[53],whichsheextendedtoalmostglobalandglobalexistence,respectively,
in [54], [55]. Independently, global existence in three dimensions was shown by
Germain, Masmoudi, and Shatah in [20].
Inthecaseoffinitedepththefirstgeneralwell-posednessresultforgravitywaves
inthree dimensions wasobtainedby Lannes[36] in2005. This resultwasextended
in[2],wherethecharacteristicdimensionlessparametersoftheoriginalwater-waves
problem have been worked out in order to provide a stable fundament for the
derivation and justification of various asymptotic limits. The results obtained, are
presented in more detail in the survey [37], to which we take explicit reference in
the presentpaper. Concerningthe well-posedness ofcapillary-gravitywater-waves,
we mention exemplarily only the more recent selection [5, 45, 24, 8, 47, 10, 4, 40,
1, 21, 38], to which we refer for more details on the various results obtained, their
development and their extensions.
As mentioned above, for each original water-wave problem with its own char-
acteristics, different asymptotic limits can be obtained. The main distinction of
the derivedmodels is with respect to the shallowness parameter µ=H2/L2 of the
0
originalequations,whereH >0 is the waterdepthandL=1 is the characteristic
0
horizontal length-scale. For µ 1 we speak of shallow water, while for µ 1 and
≪ ≈
µ > 1 of deep water (with the limiting case of infinite depth as µ ). This
→ ∞
classificationis not arbitrary. Indeed, the maindifference inthe behaviorof water-
waves in these two cases is that for increasing water depth the rˆole of dispersion
becomes more dominant, see, e.g., [37, 1.3]. Of course within each of these two
§
mainclassesofmodels,finerdistinctionscanbe,andindeedare,made. Sinceinthe
present article we consider the deep (though finite) water case µ>1, we refrain to
mentionanyofthevariousshallowwatermodels,butrefertothesurvey[37],which
seems to give a complete accountof the ”state-of-the-art”in 2013 concerning their
derivation and justification. However, we would like to mention the justification
of the celebrated KdV-equation in the two-dimensional case in [44, 45] after some
first results in [11, 33], and with improvements in [7, 25, 26].
We will address modulation equations further below, after presenting in the
following the capillary-gravitywater waves equations.
The capillary-gravity water waves problem of finite depth 0 < õ < with
∞
a flat bottom extending over all of Rd, d = 1,2, can be written in the following
non-dimensionalized form, due to Zakharov [57], Craig, C. Sulem, P.-L. Sulem
[14, 15, 49], and Alvarez-Samaniego, Lannes [2, 37]:
(1.1) ∂ U + (U)=0, U =(ζ,ψ)T, =( 1 , 2 )T,
t Nǫ,σ Nǫ,σ Nǫ,σ Nǫ,σ
INTERACTION OF MODULATED GRAVITY WATER WAVES OF FINITE DEPTH 3
where
(1.2) 1 (U)= [ǫζ]ψ,
Nǫ,σ −G
ζ ǫ ǫ( [ǫζ]ψ+ǫ ζ ψ)2
(1.3) 2 (U)=ζ 1 ∇ + ψ 2 G ∇ ·∇ .
Nǫ,σ − Bo∇· 1+ǫ2 ζ 2 2|∇ | − 2 1+ǫ2 ζ 2
(cid:16) |∇ | (cid:17) |∇ |
Here, the unknown functionspof time t [0,T), T > 0, and space X Rd are the
surface elevation ζ : [0,T) Rd R ∈and the trace ψ : [0,T) Rd∈ R of the
× → × →
velocity potential of the fluid at the surface. The scaling parameter 0 < ǫ 6 1 is
the steepness of the wave, i.e., the ratio of the amplitude of the surface elevation
above the still water level ζ =0 to the characteristic horizontal length L=1.
The secondtermin (1.3) correspondsto the surface tension,which is essentially
the mean curvature of the surface scaled by the (inverse) Bond number 1 = σ ,
Bo ρg
where σ,ρ,g are the (dimensionless) coefficients of the surface tension, the fluid
density and the gravityacceleration,respectively. When σ =0, this term is absent
and we speak of gravity water waves.
The most important term in the above formulation is the Dirichlet-Neumann
operator
(1.4) [ǫζ]ψ =∂zΦ(,ǫζ) (ǫζ) Φ(,ǫζ)= 1+ (ǫζ)2∂nΦ(,ǫζ)
G · −∇ ·∇ · |∇ | ·
where the velocity potential of the fluid Φ solves thpe boundary value problem for
the Laplace equation
∆X,zΦ=0, √µ6z 6ǫζ,
(1.5) −
(Φ(,ǫζ)=ψ, ∂zΦ(, √µ)=0
· · −
in the fluid domain at time t>0,
Ω = (X,z) Rd+1 : √µ6z 6ǫζ(t,X) ,
ǫ,t
{ ∈ − }
with Dirichlet data ψ at the surface and Neumann boundary data at the bottom.
With n in (1.4) being the upwardunit normalvector atthe surface ǫζ, we see that
the Dirichlet-Neumann operator [ǫζ] relates the Dirichlet data ψ to the normal
G
derivative of the potential Φ at the surface, thus justifying its name. In particular,
the first equation of the system (1.1),
(1.6) ∂ ζ [ǫζ]ψ =0,
t
−G
codifies the physical assumption that fluid particles at the surface stay there for
all times. We note also that the Dirichlet-Neumann operator is linear in ψ but
nonlinear in ζ.
The second equation of the gravity water-wavesproblem (1.1) (with σ =0),
ǫ ǫ( [ǫζ]ψ+ǫ ζ ψ)2
(1.7) ∂ ψ+ζ+ ψ 2 G ∇ ·∇ =0,
t 2|∇ | − 2 1+ǫ2 ζ 2
|∇ |
originates from the Euler equation for the fluid velocity (ǫΦ) of an inviscid,
X,z
∇
homogeneous, incompressible, and irrotational fluid in Ω under the influence of
ǫ,t
gravity and with constant external (atmospheric) pressure at the surface. Inte-
grating the Euler equation over the space variables (X,z) and evaluating it at the
surface z =ǫζ, one gets
1
(1.8) ∂ (ǫΦ)(,ǫζ)+ (ǫΦ)(,ǫζ)2+ǫζ =0.
t X,z
· 2|∇ · |
4 IOANNISGIANNOULIS
By use of the chain rule on ψ = Φ(,ǫζ), one obtains with (1.4) and (1.6) that
·
(1.8) is equivalent to (1.7). Thus, determining U = (ζ,ψ) via the system (1.6),
(1.7), we have all required data to solve (1.5) for Φ (under reasonable regularity
assumptionsonU andundertheconditionthattheflowisatrestas (X,z) ).
| |→∞
From the Euler equation we can then determine also the pressure of the fluid. It
was Zakharov who noted in [57] that the knowledge of U = (ζ,ψ) is sufficient for
solvingthewater-wavesprobleminthisway,whiletheuseoftheDirichlet-Neumann
operator (1.4) in the formulation of the system (1.6), (1.7) is mainly due to Craig,
C. Sulem and P.-L. Sulem in [14, 15].
The non-dimensionalized version (1.1) of the water-waves problem, that we use
forthe dispersive,deep watercaserelevantin this article,is relyingona moregen-
eral one, derived by Alvarez-Samaniego and Lannes first in [2] and then presented
in more detail in [37], which works out all characteristic parameters of the water
wave problem. This is particularly useful for a systematic and analytically reliable
derivationofallpossibleasymptoticlimitsonemaybeinterestedin. Sincetheulti-
mategoalofthepresentarticleisthejustification(seeSection4)ofthemodulation
equationsformallyderivedinSection3, andsinceourjustificationresult(Theorem
4.2) followsdirectly fromthe stability propertyof the water-wavesproblemas pre-
sentedbyLannesin[37](seehereTheorem4.1),wechosetostudythewater-waves
problem from the beginning in the form (1.1). This is also the reason for the (at
a first glance unusual) notation of the water-depth by õ. For a full derivation of
the water-wavesproblemin the form (1.1), and an extended and detailed overview
of its recent analytical state of the art, we refer the reader to [37].
The water-wavesproblem (1.1) has the linearization around (ζ,ψ)=(0,0)
∂ ζ [0]ψ =0,
t
(1.9) −G
(∂tψ+ζ− B1o∆ζ =0,
with [0]ψ = ∂ Φ(,0), where Φ solves (1.5) with Dirichlet data Φ(,0) = ψ at
z
G · ·
the surface ζ = 0. Considering the Fourier transform of (1.5) with respect to the
horizontal variables X Rd, we obtain for each ξ Rd a second-order ODE for
∈ ∈
Φ(ξ, )intheverticalvariablez withboundaryvaluesatz = √µandz =0,which
· −
can be solved uniquely, yielding
∂ Φ(ξ,0)=g (ξ)ψ(ξ), g (ξ)= ξ tanh(√µξ ), ξ Rd.
z 0 0
| | | | ∈
Thus, in the Fourier-multiplier notation
d b
(1.10) f\(D)u(ξ)=f(ξ)u(ξ), ξ Rd, with D = i ,
∈ −∇
we obtain
b
[0]ψ = D tanh(√µD )ψ.
G | | | |
Moreover,we obtain that (1.9) allows for plane wave solutions of the form
ζ
(1.11) ei(ξX ωt)+c.c., ξ Rd, ω R, ζ,ψ C
ψ · − ∈ ∈ ∈
(cid:18) (cid:19)
(with c.c. denoting the complex conjugate of the preceding term(s)), provided the
dispersion relation
iω
(1.12) ζ = ψ and ω2 =ω2(ξ),
1+ 1 ξ 2
Bo| |
INTERACTION OF MODULATED GRAVITY WATER WAVES OF FINITE DEPTH 5
is satisfied, with the dispersion function
(1.13) ω(ξ)= (1+ 1 ξ 2)g (ξ), g (ξ)= ξ tanh(√µξ ), ξ Rd.
Bo| | 0 0 | | | | ∈
q
Inthecaseoflinearsystemsonecanconstructmorecomplicatedsolutions(wave
packets) by superposition of plane waves via Fourier transformation. The analog
to this in nonlinear systems is the consideration of modulated plane waves. In the
most simple case of amplitude modulation we replace the constants ζ,ψ C in
∈
(1.11) by slowly varying functions
ζ(t,X )
(1.14) ′ ′ ei(ξX ωt)+c.c., ζ,ψ :[0, ) Rd C,
ψ(t′,X′) · − ∞ × →
(cid:18) (cid:19)
where t =ǫt, X =ǫX with 0<ǫ61 are new, macroscopictime- and space-vari-
′ ′
ables. The question then is whether the nonlinear system allows approximatively
forsolutionsofthisform. Bythiswemeansolutionswhichmaintain theaboveform
at least locally with respect to the macroscopic time t 6 T, or, equivalently, for
′
t6T/ǫ. Typically, by inserting the two-scale ansatz (1.14) into the nonlinear sys-
tem, one obtains formally the necessary conditions, viz. the modulation equations,
whichthe macroscopicfunctionsζ,ψ havetosatisfy. Themodulationequationsre-
veal some qualitative, macroscopic feature in the behavior of the nonlinear system
under investigation, which depends of course strongly on the macroscopic scaling
usedfor the modulation. This approachhasbeen usedwidely inthe physicslitera-
tureforallsortsofdispersivesystems. Indeed,oneoftheoldestfieldsofapplication
havebeen waterwaves,asis exemplifiedprominently inthe workofWhitham[51],
to which we refer for a methodical exposition of the ideas behind modulation from
the physical point of view.
Innonlineardispersivesystems the centralmodulationequationisthe nonlinear
Schr¨odingerequation(nlS),sinceitcapturestheinterplaybetweennonlinearityand
dispersiongoverningthe deformation of the envelopes of the wave packets (see e.g.
[49] for an overview). For this, the right two-scale ansatz is not (1.14) but rather
ζ(t ,X )
(1.15) ′′ ′′ ei(ξX ωt)+c.c., ζ,ψ :[0, ) Rd C,
ψ(t′′,X′′) · − ∞ × →
(cid:18) (cid:19)
witht =ǫt, X =X ωt,where ω isthe groupvelocityofthe wavepacket.
′′ ′ ′′ ′ ′
−∇ ∇
Inserting this ansatz (with a correspondingpolarizationconditionfor ζ) into (1.1),
one obtains the nlS equation, which for two-dimensional gravity waves takes the
form
1
∂t′′ψ−i2ω′′∂x′′2ψ+ic|ψ|2ψ =0
with c R depending on ξ,ω and ∂ , ∂ denoting differentiation with respect to
∈ t′′ x′′
t , x . It was derived by Zakharov [57] for infinite depth and by Hasimoto and
′′ ′′
Ono [30] for finite depth. In the three-dimensional case of finite depth instead of
the nlS one obtains for the scaling (1.15) the Davey-Stewartson (DS) system [16]
(see[49]foradetaileddiscussionofitsproperties). However,ininfinitedepthagain
the nlS is obtained as the modulation equation for the scaling (1.15). Concerning
the justification of these modulation equations, this has been achievedfor the two-
dimensionalgravitywater-wavesproblembyTotzandWu[50]inthecaseofinfinite
depth and by Du¨ll, Schneider, and Wayne [18] in the case of finite depth. In the
three-dimensional capillary-gravity case there exist consistency results for the nlS
6 IOANNISGIANNOULIS
equation[12]andfortheDSsystem[13],whereconsistencymeansthattheamount
by which the approximate solution fails to satisfy the original problem (i.e. the
residual)tends to zeroin the asymptotic limit with respect to some relevantnorm.
In the present article we use the hyperbolic scaling (1.14) and consider three
modulatedplanewavesofthatform. Weareinterestedinthemodulationequations
thatgovernthe macroscopicdynamics ofthese wavesnotonly inleading orderbut
also with respect to their macroscopic corrections of order O(ǫ). For the sake of
clarity, we first present our exact assumptions, and discuss them afterwards.
We make the two-scale ansatz for approximate solutions of (1.1)
ζ ζ ζ ζ
(1.16) U = a = 0 +ǫ 1 +ǫ2 2
a ψ ψ ψ ψ
a 0 1 2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
with
ζ ζ ζ
0 = 0j e +c.c.+ 00 ,
ψ ψ j ψ
0 0j 00
(cid:18) (cid:19) j (cid:18) (cid:19) (cid:18) (cid:19)
X
ζ ζ ζ ζ
1 = 1j e + 1ji e +c.c.+ 10 ,
ψ ψ j ψ ji ψ
1 1j 1ji 10
(cid:18) (cid:19) j (cid:18) (cid:19) ji (cid:18) (cid:19) (cid:18) (cid:19)
X X
ζ ζ ζ ζ ζ
2 = 2j e + 2ji e + 2jik e +c.c.+ 20 ,
ψ ψ j ψ ji ψ jik ψ
2 2j 2ji 2jik 20
(cid:18) (cid:19) j (cid:18) (cid:19) ji (cid:18) (cid:19) jik (cid:18) (cid:19) (cid:18) (cid:19)
X X X
defined according to the following notations and assumptions:
Notation 1.1. (1) All functions ζ ,ψ on the right of (ζ ,ψ )T, n = 0,1,2,
... ... n n
are complex-valued and depend only on the macroscopic time- and space-variables
06t =ǫt6T, X =ǫX =ǫ(x,y) Rd (d=1,2 and X =x R if d=1), where
′ ′
∈ ∈
0 < ǫ 6 1. Differentiation with respect to t and X is denoted by ∂ and . The
′ ′ t′ ∇′
abbreviation c.c. denotes the complex conjugate of all preceding terms.
(2) We introduce the index-sets
J = 1,2,3 ,
{ }
I = (1,1), (2,2), (3,3), (1, 2), (1, 3), (2, 3) (1,2),(1,3),(2,3) =I ,
<
{ ± ± ± }⊃{ }
K = (1,1,1), (2,2,2), (3,3,3),
{
(1,1, 2), (1,1, 3), (2,2, 3),(2,2, 1), (3,3, 1), (3,3, 2),
± ± ± ± ± ±
(1,2,3), (1,2, 3), (1,3, 2), (2,3, 1) .
− − − }
We denote summation over these index-sets by
:= , := , := .
Xj Xj∈J Xji (jX,i)∈I Xjik (j,iX,k)∈K
(3a) The functions e for j J, e for (j,i) I and e for (j,i,k) K are
j ji jik
± ∈ ∈ ∈
defined through
e j(t,X)= e±i(ξj·X−ωjt), eji = ejei, ejik = ejeiek,
±
INTERACTION OF MODULATED GRAVITY WATER WAVES OF FINITE DEPTH 7
where the wave-vectors ξ Rd 0 and the frequencies ω =ω(ξ )>0 satisfy for
j j j
∈ \{ }
each j J the dispersion relation ω2 =ω2(ξ ) with the dispersion function (1.13).
∈ j j
We assume that the plane waves e , j J, are mutually different, i.e.
j
∈
(ξ ,ω )=(ξ ,ω ) j,i J, j =i.
j j i i
6 ∀ ∈ 6
Occasionally, we will refer to e , e and e as the first-, second- and third-order
j ji jik
harmonics, respectively, and to 1= e0 as the zeroth-order harmonic.
(3b) In analogy to the index-notation for the harmonics, we use the abbreviations
ξ = ξ , ξ =ξ +ξ , ξ =ξ +ξ +ξ ,
j j ji j i jik j i k
± ±
ω = ω , ω =ω +ω , ω =ω +ω +ω ,
j j ji j i jik j i k
± ±
and
b =1+ 1 ξ 2, b =1+ 1 ξ 2, b =1+ 1 ξ 2,
j Bo| j| ji Bo| ji| jik Bo| jik|
g =g (ξ ), g =g (ξ ), g =g (ξ ).
j 0 j ji 0 ji jik 0 jik
Finally, we denote
g = g (ξ ), g = g (ξ ), H = 1 (ξ ) ,
j′ ∇ 0 j j′i ∇ 0 ji j 2∇′·Hg0 j ∇′
where (ξ) is the Hessian matrix of the function g at ξ Rd, see (1.13).
Hg0 0 ∈
(Note, in particular, g0(0,0)=0, ∇g0(0,0)=(0,0) and Hg0(0,0)=2√µI.)
In this notation the following identities hold true:
(1.17) 2ω ω =b g + 1 2ξ g , where ω = ω(ξ ),
j∇ j j j′ Bo j j ∇ j ∇ j
and
b 1 ω
(1.18) (ξ ) ψ = j H ψ (b ω )2ψ + 1 j∆ψ ,
∇′·Hω j ∇′ 0j ω j 0j − ω j∇Bo j ·∇′ 0j Bo b ′ 0j
j j j
where
1 ω
(1.19) ω = ω 1 2 jξ .
∇Bo j b ∇ j − Bo b j
j j
(cid:16) (cid:17)
(3c) The plane waves e , j J, satisfy the non-resonance conditions
j
∈
ω2 =ω2(ξ )=b g (j,i) I
ji 6 ji ji ji ∀ ∈
and
ω2 =ω2(ξ )=b g (j,i,k) K.
jik 6 jik jik jik ∀ ∈
(4) For u Hs(Rd), s R, see (4.3), of the form
∈ ∈
k
u(X)= u˜i(X′)eiξi·X
i=1
X
we use the notations u˜ = k u˜ and
| |Hs i=1| i|Hs
k P k
u′(X)= iξiu˜i(ǫX)eiξi·X, u′′(X)= ( ξi 2)u˜i(ǫX)eiξi·X,
−| |
i=1 i=1
X X
such that (with ∆ = )
′ ′ ′
∇ ·∇
(1.20) u=u +ǫ u and ∆u=u +2ǫ u +ǫ2∆u.
′ ′ ′′ ′ ′ ′
∇ ∇ ∇ ·
8 IOANNISGIANNOULIS
Themotivationforthespecialformoftheansatz(1.16)isthatwewanttoinclude
inourformalexpansionofthecapillary-gravitywater-wavesequation(1.1)thecase
ofquadraticinteractionoftwomodulatedplanewaves. Byquadraticinteractionwe
mean the situation where two such waves generate by multiplication a third plane
wave through the (quadratic) resonance of their phases, e.g. e e = e . In this
1 2 3
case one has to consider from the outset all three involved modulated plane waves
in order to obtain a closed system of macroscopic equations, and the interaction is
manifested macroscopicallyin leading order,which means that an expansionup to
O(ǫ)-terms in (1.16) would be sufficient.
However, as will be explained below, in the case of pure gravity waves, which
is our main focus in the present paper, no such quadratic resonances arise. Such
resonancesexistonlyifsurfacetensionisincludedintheoriginalwater-wavesequa-
tion, see [46] for the two-dimensional case (d = 1). Then, naturally, the question
arises, whether even in this quadratically non-resonant case any macroscopic cou-
pling can be detected in the next-to-leading-order correction of the leading order
amplitudes or in the non-oscillating mean field generated by the waves. Wanting
to perform the (unsurprisingly, very cumbersome) formal expansion of the water
waves equation for three modulated pulses only once, we chose the ansatz (1.16),
which is usefull in both cases (quadratically resonant and non-resonant) and for
waves with or without surface tension.
As expected, indeed also in the quadratically non-resonantcase, the interaction
of modulated waves can be traced in the second-order macroscopic system. More
precisely,weobtaininSection3thatinorderfortheapproximationU of (1.16)to
a
satisfyformallythe waterwavesequation(1.1)uptoresidualtermsoforderO(ǫ3),
i.e.
(1.21) ∂ U + (U )=ǫ3(r1,r2)T,
t a Nǫ,σ a 2 2
the macroscopic modulation equations
∂ ψ + ω ψ =0,
t′ 0j ∇ j ·∇′ 0j
(3.17) ∂∂tt′′ψ2ψ1j00+−∇√ωµj∆·∇′ψ′0ψ01j==PEjj(cid:16)(gj2−|ξj|2)∂t′+2ωbjjξj ·∇′(cid:17)|ψ0j|2,
with j J = 1,2,3 and
∈ { }
1 b
(3.9) Ej = i2∇′·Hω(ξj)∇′ψ0j −iψ0j 2ωj (gj2−|ξj|2)∂t′+ξj ·∇′ ψ00+E˜j
j
(cid:16) (cid:17)
have to be satisfied, where E˜ consists of cubic products of the leading-order am-
j
plitudes ψ , see (3.10). The other macroscopic functions appearing in U can be
0j a
determined via ψ ,ψ ,ψ or can be chosen arbitrarily.
0j 00 1j
Ofcourse,moregenerallyonecouldconsideralsoanarbitrarynumberofN N,
∈
N >3, modulated plane waves,as was done for other physical settings, e.g. in [23,
22]. Inorderto keepthe presentationmoresimple andexplicit, we chose,however,
to consider here only three pulses. Note that the results derived in the present
paper can be used in order to obtain the (non-resonant) macroscopic dynamics up
to next-to-leading order for two pulses or even for a single pulse, by equating the
macroscopiccoefficientsofthe superfluouswavesto zero,cf. fortwo pulses e.g.[29]
and for a single pulse [6] (see also Remark 3.3) and [49, 11.1], [37, 8.2.5].
§ §
INTERACTION OF MODULATED GRAVITY WATER WAVES OF FINITE DEPTH 9
Theansatz(1.16)consistsofallfirst-andhigher-orderharmonictermsexpected
to arise up to order ǫ2 due to the nonlinear nature of the water wavesproblem, in-
cludingthenon-oscillatingtermsforthemeanfieldwhicharisefromtheinteraction
of a plane wave with its complex conjugate. Note here, that the index sets I and
K of Notation 1.1(2) represent all possibly different second- and third-order har-
monics. In this context, we point out that actually we are interested in the weakly
nonlinear approximation of solutions to the water-wavesproblem (1.1) with ǫ=1,
that is to say in an approximation of the form ǫU with U as in (1.16). However,
a a
the expansion of the water waves equation in Section 2 and the derivation of the
modulationequationsinSection 3areperformedfor the equationinthe form(1.1)
and the ansatz U in (1.16).
a
The reason why the ansatz U includes also terms of order ǫ2, although we are
a
interested only in an approximation U of up to next-to-leading-order terms of
a,1
order ǫ (i.e. consisting only of the first two terms of U ), is that the determining
a
equationsforthefunctionsζ ,ψ ariseatorderO(ǫ2),cf.Corollary2.5. Moreover,
1j 1j
for the justification of the approximation U over time scales of order O(1/ǫ) we
a,1
need first to consider an approximation that satisfies (1.1) formally up to residual
terms of order O(ǫ3), see Section 4 for the details, and [34] for a more general
presentation of this approach.
The two non-resonance conditions of Notation 1.1(3c) imply that none of the
higher-order harmonic terms are plane waves (or non-oscillating). The first set of
conditionsisessentialforthe formofthemodulationequations,see(3.17),yielding
thatinleadingorderthe macroscopicamplitudes ψ arejust transportedwiththe
0j
group velocity ω of the wave packet, without any macroscopic interaction. In
j
∇
the opposite case, when quadratic resonances appear, one obtains the ’three-wave-
interaction equations’, a coupled system of three semilinear transport equations
for ψ containing for each j J quadratic products of the other two amplitudes,
0j
∈
according to the existing resonances, see [46]. As stated above, for pure gravity
water waves this quadratic non-resonance condition is always satisfied. This is
known since the 1960s, see [42], while the existence of quadratic resonances in the
caseofcapillary-gravitywavesofinfinite depthwas firstprovenin[39]. For amore
generaldiscussionofresonancesofwaterwaveswereferto[28,46]andthereferences
given therein. For the sake of completeness we give in the following Remark 1.1 a
shortanalyticalproofofthenon-existenceofquadraticresonancesforgravitywater
waves of finite depth.
Remark 1.1. Assume there are ξ ,ξ ,ξ +ξ Rd 0 , such that
1 2 1 2
∈ \{ }
(ω(ξ ) ω(ξ ))2 =ω2(ξ +ξ ) with ω(ξ)= ξ tanh(√µξ ).
1 2 1 2
± | | | |
q
Sinceω is aradialfunction, takingsquare-rootsandpossiblyconsideringthe oppo-
site of some wave-vector ξ and relabeling, the equation on the left can always be
j
written in the form
ω(ξ )+ω(ξ )=ω(ξ +ξ ), ξ ,ξ Rd 0 .
1 2 1 2 1 2
∈ \{ }
1
Mutiplying by µ4 >0 and setting
ξ ξ ξ
2 1 2
x=√µξ >0, λ= | | >0, c= · [ 1,1],
1
| | ξ ξ ξ ∈ −
1 1 2
| | | || |
10 IOANNISGIANNOULIS
solving this equation for ξ ,ξ Rd 0 is equivalent to finding roots (x,λ,c)
1 2
∈ \{ } ∈
(0, ) (0, ) [ 1,1] of the function
∞ × ∞ × −
r (x,λ,c)= (1+λ2+2cλ)tanh(x 1+λ2+2cλ) λtanh(xλ) √tanhx.
0
− −
q
But r (x,λ,c)6r (x,λ,1) and p p
0 0
g(x(1+λ)) g(xλ)
r (x,λ,1)= − √tanhx, where g(y)= ytanhy,
0
√x −
p
with r (x,0,1)=0 and
0
d
r (x,λ,1)=√x(g (x(1+λ)) g (xλ))<0 λ>0,
0 ′ ′
dλ − ∀
the latter due to the strict decreasing of
tanhy+y(1 tanh2y)
g′(y)= − , y >0.
2√ytanhy
Hence, we conclude r (x,λ,c)<0 for all (x,λ,c) (0, ) (0, ) [ 1,1]. (cid:3)
0
∈ ∞ × ∞ × −
While the first (quadratic) non-resonance condition of Notation 1.1(3c) is es-
sential for the form of the derived modulation equations, the second (cubic) non-
resonance condition is much less so. Indeed, if the first non-resonance condition
holds true, cubic resonances do not change the form of the modulation equations,
but merely contribute additional cubic products of the leading order amplitudes
ψ to the source term of one of the equations for the next-to-leading order ampli-
0j
tudes ψ , if a third-order harmonic is equal to one of the three considered plane
1j
waves. (Note, that in general if plane waves are generated through resonant inter-
action one always has to include the generated wave in the original ansatz, here
(1.16),inordertoobtainaclosedsystemofmodulationequations.) Forthe sakeof
simplicity we do not consider these cases explicitly here, but prefer to impose the
cubic non-resonance condition of Notation 1.1(3c) instead. Nevertheless, since the
same justification result holds true for these modified modulation equations, the
presentpapercoverscompletelythe justificationofthemodulationequationsupto
next-to-leading order for three weakly amplitude-modulated gravity water waves,
provided possibly existing cubic resonances generate only one of the three plane
waves considered.
We close this introduction by outlining the structure of the article and com-
menting on its main results. In the following Section 2, after inserting the ansatz
(1.16) for the approximation U into the water waves equation (1.1), we expand
a
with respect to the steepness parameter 0 < ǫ 1 up to residual terms of formal
≪
order O(ǫ3), for which we give estimates in the Sobolev norms used for the justifi-
cationofthemodulationequations(3.17)inSection4. Thepreciseformulasforthe
more involved,though structurally simple, macroscopic coefficients are given in an
Appendix. Thus, as a byproduct, we provide a complete formal explicit expansion
includingalltermsoforderε2 for threeamplitude modulatedplanewaveswiththe
hyperbolic scaling t = ǫt, X = ǫX for the capillary-gravity water waves problem
′ ′
of finite depth, that can be used independently, also for only one or two waves.
Then, in Section 3, we derive the necessary conditions on the macroscopic co-
efficients of U in order for the latter to satisfy (1.1) up to the residual terms of
a