EFFECTIVE BEHAVIOR OF SOLITARY WAVES OVER RANDOM TOPOGRAPHY

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M ULTISCALE M ODEL.S IMUL.c 2007Society for Industrial and Applied Mathematics Vol.6,No.3,pp.995–1025

EFFECTIVE BEHA VIOR OF SOLITARY W A VES OVER RANDOM

TOPOGRAPHY?

JOSSELIN GARNIER?,JUAN CARLOS MU?NOZ GRAJALES?,AND ANDR′E NACHBIN§

Abstract.The deformation of a nonlinear pulse traveling in a dispersive random medium can be studied with asymptotic analysis based on separation of scales when the propagation distance is large compared to the correlation length of the random medium.We consider shallow water waves with a spatially random depth.We use a formulation in terms of a terrain-following Boussinesq system. We compute the e?ective evolution equation for the front pulse which can be written as a dissipative Kortweg-de Vries equation.We study the soliton dynamics driven by this system.We show,both theoretically and numerically,that a solitary wave is more robust than a linear wave in the early steps of the propagation.However,it eventually decays much faster after a critical distance corresponding to the loss of about half of its initial amplitude.We also perform an asymptotic analysis for a class of random bottom topographies.A universal behavior is captured through the asymptotic analysis of the metric term for the corresponding change to terrain-following coordinates.Within this class we characterize the e?ective height for highly disordered topographies.The probabilistic asymptotic results are illustrated by performing Monte Carlo simulations with a Schwarz–Christo?el Toolbox.

Key words.nonlinear waves,water waves,random media,solitons

AMS subject classi?cations.76B15,35R60,35Q53

DOI.10.1137/060676064

1.Introduction.We address the propagation of nonlinear dispersive free sur-face waves in a disordered one-dimensional?uid body.Regarding wave-topography interaction in a homogeneous medium,three characteristic length scales are impor-tant in studying di?erent regimes of propagation:the typical depth h0,the typical wavelengthλ0,and the typical amplitude of the wave elevation a0.We consider the framework corresponding to shallow water waves where a0 h0 λ,so that the problem can be written as a weakly dispersive,weakly nonlinear system.In this paper the lower boundary is a disordered surface modeled by a stationary random process. Two new length scales appear:the horizontal length scale,de?ned as the correlation length of the random?uctuations of the bottom,is denoted by l c,while the typical amplitude of the random?uctuations is denoted byδh.We shall study this problem in two di?erent asymptotic regimes.We assume either that the amplitude of the ?uctuations of the bottom is small compared to the average depthδh h0,or that the typical amplitude is of the same order as the average depthδh～h0,but the cor-relation length is much smaller than the typical wavelength l c λ.We also assume that the propagation distance is large.We carry out an asymptotic analysis based on these assumptions.Our goal is to derive an e?ective evolution equation governing

?Received by the editors November27,2006;accepted for publication(in revised form)August27, 2007;published electronically December19,2007.

815cedc0d5bbfd0a79567398/journals/mms/6-3/67606

?Laboratoire de Probabilit′e s et Mod`e les Al′e atoires&Laboratoire Jacques-Louis Lions,Universit′e Paris7,2Place Jussieu,75251Paris Cedex05,France(garnier@math.jussieu.fr).This author’s research was supported by ANR.

?Departamento de Matem′a ticas,Universidad del Valle,A.A.25360Cali,Colombia(jcarlmz@ 815cedc0d5bbfd0a79567398).This author’s research was supported by COLCIENCIAS under grant11060516858.

§Instituto de Matem′a tica Pura e Aplicada,Est.D Castorina110,Jardim Bot?a nico,Rio de Janeiro, RJ22460-320,Brazil(nachbin@impa.br).This author’s research was supported by CNPq/Brazil under grant300368/96-8.

995

996J.GARNIER,J.C.MU?NOZ GRAJALES,AND A.NACHBIN

the propagation of the free surface wave.This is done by applying an asymptotic sto-chastic analysis to the Lagrangian formulation of the problem,following the strategy that we introduced in[12]in a weakly heterogeneous regime for a nonlinear hyper-bolic system.Here we start with a one-dimensional,shallow water Boussinesq system which is transformed into a Lagrangian frame by using the Riemann invariants of the underlying nondispersive,constant coe?cient system.Applying a limit theorem for stochastic di?erential equations we characterize the?ow along the wavefront by a viscous Kortweg–de Vries(KdV)equation.Actually the di?usive-like term is more complicated than an e?ective viscosity and has the form of a pseudodi?erential op-erator,but it reduces to such a term when the correlation length of the medium is much smaller than the pulse width.

We give a brief background on the mathematical theory that has been developed and which is along the lines of our study.We use a mathematical formulation that has been successfully used by the authors in other nonlinear settings,namely a nonlinear conservation law(shallow water system)and a nonlinear advection di?usion equation (viscous shallow water model).Now we apply the technique to a di?erent class of partial di?erential equations(PDEs).In what follows we use Riemann invariants in a weakly dispersive,weakly nonlinear setting for which we can consider solitary waves interacting with disorder.We shall contrast the behavior of a solitary wave with its linear counterpart.The propagation of a linear pulse through a random medium has been extensively studied[1].In particular the O’Doherty–Anstey(ODA) theory predicts that if the pulse is observed in a Lagrangian frame that moves with a random velocity,then the pulse appears to retain its shape up to a slow spreading and attenuation[31].A rather convincing heuristic explanation of this phenomenon is given in[5].The mathematical treatment of this issue is addressed in[5,6,20,21,3]. An extension to dispersive water waves is provided in[11,13,25].We have extended this theory to inviscid nonlinear waves in[12]and to viscous waves in[16].It is also possible to apply a mean-?eld approach to water waves over a rough bottom[22,23].

The paper is organized as follows.In section2we introduce the nonlinear shallow water wave model with a random depth together with the corresponding Riemann invariants.In section3we derive the e?ective viscous KdV equation governing the evolution of the front pulse in the case of a small-amplitude slowly varying topography. Section4is devoted to the same problem with a rapidly varying topography.In this section we also perform an asymptotic analysis for a class of rapidly varying bottom topographies.Within this class,a universal behavior is captured through the asymptotic analysis of the metric term for the corresponding change to terrain-following coordinates.In section5we discuss and compare the pulse attenuation and spreading in the linear regime and in the soliton regime.In section6we present numerical simulations to illustrate the accuracy of the theoretical predictions of the asymptotic analysis.

2.Shallow water waves with random depth.

2.1.The terrain-following Boussinesq system.The water wave problem addressed here is in a regime where Euler’s equations are valid.At the top of our domain a free surface de?nes the wave pro?le.Euler’s equations can then be recast as a potential theory problem where(?u,?v)=?φ(x,y,t).Thus the velocity?eld is obtained through the velocity potentialφ[36].In[29]a conformal map is used to map the rough channel onto a?at strip.This is the same as changing variables from a Cartesian xy-coordinate system to an orthogonal curvilinear one,namely,in the ξζ-variables.This is done in such a way that at the undisturbed free surface y≡0 Copyright ? by SIAM. Unauthorized reproduction of this article is prohibited.

Copyright ? by SIAM. Unauthorized reproduction of this article is prohibited. SOLITARY WAVES OVER RANDOM TOPOGRAPHY 997and ζ≡0coincide.The topography now is along a (ζ≡constant)curve.In [29]this constant is such that (ζ≡√β).Since we are dealing with harmonic functions (through the potential and the conformal map)it easily follows that in the curvilinear coordinates we have a new set of orthogonal velocity components (u,v )=?φ(ξ,ζ,t ),where u (ξ,ζ,t )is a terrain-following component,namely,tangent to the ζ-level curves.Now the gradient is to be considered in the new set of variables.It follows [27]that a

family of Boussinesq systems can be deduced,depending on the depth ζat which the

u -velocity component is evaluated.As can be seen in [27]this changes the dispersion relation in di?erent interesting ways,through a family of Pad′e approximations from the original potential theory dispersion relation.By an asymptotic analysis along the depth,the two-dimensional Euler equations can be reduced to a one-dimensional system for the wave elevation η(ξ,t )and the

terrain-following velocity component u (ξ,ζ,t ),where ζis an arbitrary ?xed depth [27,29,36].As our reduced model of interest,we consider the one-parameter fam-ily of Boussinesq equations that describe the evolution of surface waves in shallow channels [27]Mηt + 1+αηM u ξ?β2 y 20?13 [Mη]ξξt =0,(2.1)u t +ηξ+α u 22M 2 ξ+β2(y 20?1)u ξξt =0,(2.2)where ηis the wave elevation and u is the terrain-following velocity at a relative depth parameterized by y 0[29,27]in such a way that y 0=0is the bottom and y 0=1is the free surface.ξand t are the space and time coordinates,respectively.The parameter αis the ratio of the typical wave amplitude over the mean depth.It governs the strength of the nonlinearity.The parameter βis the ratio of the squared mean depth over the squared characteristic wavelength of the wave.It governs the strength of the dispersion.These two parameters are assumed to be small.The system (2.1)–(2.2)is derived from ?rst principles (mass and momentum conservations)in the asymptotics

α 1and β 1,and it neglects terms of order O (α2),O (αβ),and O (β2).Hence,as mentioned above,the system is a weakly nonlinear,weakly dispersive asymptotic

approximation of the potential theory equations for an irrotational,incompressible,and inviscid ?uid.For example,this is the regime where one might have tsunamis.At

early stages these waves have a few meters of height and tens of kilometers of length.Hence we are in the linear regime,and as they approach the continental shelf we enter the weakly nonlinear regime,which will account for shoaling and steepening [36].

Moreover,this long wave,of tens of kilometers or more,propagates over regions of thousands of meters and less.Hence it is clear that in this scenario α 1and β 1

are representative.Moreover,the disordered microscale features of the topography,which play a role within the framework of this paper,are those in the range of hundreds of meters to a few kilometers.A ?rst version of this terrain-following model was given in [29],where the terrain-following velocity is depth-averaged in ζ.Then it was shown in [27]that depth-averaging u is exactly the same as using y 20=1/3in the equation

above.Along this frame of ideas,the model in [29]was then modi?ed in [35],to the same order of approximation,having as a special feature the fact that the full existence and well-posedness proof could be established.A property that played a key role in the proof was that for y 20=2/3an energy-type integral could be found.Here,for a di?erent reason,the same parameter value will play an important role.

The variable coe?cient M (ξ)is a smooth orography-dependent function which

Copyright ? by SIAM. Unauthorized reproduction of this article is prohibited. 998J.GARNIER,J.C.MU ?NOZ GRAJALES,AND A.NACHBIN appears as a consequence of a change of variables from Cartesian to curvilinear co-

ordinates.M (ξ)is the leading-order term of the Jacobian of the transform,and it is deduced from the physical orography y =h (x )which describes the channel’s depth,whose averaged value has been normalized to 1.The topography pro?le h (x )=1+n (x )can be rapidly varying,discontinuous,or even multivalued,and no mild slope condition is required.The only requirement is that there exists a con-stant C ∈(0,1)such that n ∞≤C .In the terrain-following system the physical orography h (x )=1+n (x )is replaced by the metric coe?cient (2.3)M (ξ)=1+π4√β ∞?∞n (x (ξ0,?√β))cosh 2 π2√β(ξ0

?ξ) dξ0,where (x,y )→(ξ,ζ)is the coordinate transform used for the conformal mapping in the derivation of the system [29].Note that the amplitude of M can be of order 1and it is a C ∞-function by the convolution with the sech 2function.In this paper we shall model the random topography n (x )as the realization of a

stationary random process.This in turn implies that M is a smooth random process.The random process n that describes the ?uctuations of the bottom is assumed to be bounded by a deterministic constant less than 1and to have strong mixing properties.The autocorrelation function (2.4)γ(x )=E [n (y )n (y +x )]is assumed to decay fast enough so that it belongs to L 1/2;i.e.,γdecays at in?nity fast enough to ensure the convergence of the integral ∞?∞|γ(x )|1/2dx .We de?ne the

correlation length of the medium as (2.5)l c = ∞?∞|γ(x )|dx γ(0).It represents the typical variation length scale of the random topography.2.2.The Riemann invariants.We introduce the local propagation speed cor-responding to the ?at bottom c =√1+αη.We can reformulate (2.1)–(2.2)in terms of c and u to obtain c t +α2c 1+c 2?1M u ξ+ c 2?1M ξu ?β2c y 20?13 (cc t )ξξ=0,(2.6)u t +α u 2 ξ+1 c 2?1 ξ+β(y 20?1)u ξξt =0.(2.7)We de?ne the Riemann invariants (corresponding to the unperturbed nonlinear hy-perbolic system):(2.8)A (ξ,t )=αu ?2c +2α,B (ξ,t )=αu +2c ?2α.Indeed,if the dispersion parameter is vanishing β=0and the bottom is ?at M =1,then we get back the standard left-and right-going modes (A and B ,respectively)of the nonlinear hyperbolic system:A t ?c ?A ξ=0,B t +c +B ξ=0,

SOLITARY WAVES OVER RANDOM TOPOGRAPHY999 with c?=c?αu=1?α(3A+B)/4and c+=c+αu=1+α(A+3B)/4.The identities(2.8)can be inverted:

u=A+B

,c=1+α

B?A

Substituting these expressions into(2.6)–(2.7),we get the system governing the dy-namics of the Riemann invariants in the presence of nonlinearity,dispersion,and randomness.If we neglect the terms of orderα2,the Riemann invariants satisfy

A t?Aξ+α

(3A+B)Aξ?

Aξξt=

?y20

Bξξt

(Aξ?Bξ)+

(A?B)

+αAAξ

(A?B)(Aξ?Bξ)

?α

(A?B)2+

(3A2+2AB?B2)

(2.9)

B t+Bξ+α

(3B+A)Bξ?

Bξξt=

?y20

Aξξt

(Aξ?Bξ)+

(A?B)

+αBBξ

(A?B)(Aξ?Bξ)

?α16

(A?B)2+

(?A2+2AB+3B2)

(2.10)

In these equations we have neglected terms of order O(α2),consistently with the derivation of the system(2.1)–(2.2),but we have kept all terms depending on M. In the absence of random perturbations,that is,if M=1,these equations can be reduced to

A t?Aξ+α

(3A+B)Aξ?

Aξξξ=

?y20

Bξξt,

(2.11)

B t+Bξ+α

(3B+A)Bξ+

Bξξξ=

?y20

Aξξt.

(2.12)

By choosing y20=2/3,the right-hand sides vanish,and we get that the system supports pure left-and right-going waves satisfying a KdV equation.From now on we adopt this choice for y0.Recall that this is the special value that enables the existence and well-posedness proof given in[35].

In the following sections,we consider the system(2.9)–(2.10)in the presence of a random topography in the right half-space,while the medium is unperturbed in the left half-space.The system is completed by the initial condition corresponding to a right-going wave incoming from the homogeneous left half-space

(2.13)A(ξ,t)=0,B(ξ,t)=f(t?ξ),t<0,

where the function f is compactly supported in(0,∞).

1000J.GARNIER,J.C.MU?NOZ GRAJALES,AND A.NACHBIN

2.3.The linear hyperbolic approximation.If we neglect terms of orderαandβ,that is,if we neglect all nonlinear and dispersive contributions,then the system for the Riemann invariants can be reduced to

1 2

1+1

1?1

?1+1

?1?1

?ξ

?11

The matrix in front of the partialξ-derivative can be inverted,which gives

(2.14)?

?ξ

Mξ

1?1

?11

where

(2.15)Q(ξ)=1

M(ξ)+1M(ξ)?1

1?M(ξ)?M(ξ)?1

The identity(2.14),which holds true up to terms of order O(α),O(β),will be used in the forthcoming sections to rewrite the system(2.9)–(2.10)for the Riemann invariants as a PDE of the form

??ξ

=F(A,B,A t,B t,A tt,B tt,A ttt,B ttt,M,Mξ,Mξξ),

with the same accuracy as the original system,that is,up to terms of order O(α2), O(β2),and O(αβ).

In the linear hyperbolic approximation,(2.14)can be processed directly to study the wave dynamics.Indeed,the matrix Q can be diagonalized.The eigenvalues of the matrix Q(ξ)are±M1/2(ξ).We introduce the matrix U de?ned by

(2.16)U(ξ)=1

M1/4(ξ)+M?1/4(ξ)?M1/4(ξ)+M?1/4(ξ)

?M1/4(ξ)+M?1/4(ξ)M1/4(ξ)+M?1/4(ξ)

which is such that

U?1(ξ)Q(ξ)U(ξ)=M1/2(ξ)

0?1

In terms of the new variables

(ξ,t)=U?1(ξ)

(ξ,t)

M1/2(ξ)

(2.14)has the simple form

(2.17)?

?ξ

=M1/2

0?1

Mξ

This equation clearly exhibits the two relevant phenomena in linear random medium. The?rst term in the right-hand side describes a change of the velocity described by M1/2.The second term in the right-hand side describes a coupling between the two modes imposed by the term Mξ/M,which is the re?ectivity coe?cient as in[3].

Copyright ? by SIAM. Unauthorized reproduction of this article is prohibited. SOLITARY WAVES OVER RANDOM TOPOGRAPHY 10013.Small-amplitude,slowly varying topography.We consider the weakly nonlinear,weakly dispersive system (2.9)–(2.10).We introduce a small dimensionless parameter ε>0so that the small nonlinear and dispersion parameters can be written as (3.1)α=ε2α0,β=ε2β0.Here α0(resp.,β0)is the normalized nonlinearity (resp.,dispersion)parameter which is a nonnegative number of order 1.In this section we also assume that the random topography has small-amplitude ?uctuations and is slowly varying ,in the sense that its correlation length is of the same order as the typical wavelength of the incoming

wave.It turns out that the suitable scaling that gives rise to a nontrivial asymptotic regime consists in taking (3.2)n (x )=εn 0(x ),where n 0is a stationary random process with standard deviation and correlation length of order one.3.1.Smooth topography.In this section we assume that n 0possesses deriva-tives and that it satis?es the moment conditions E [n 0(x 0)2]<∞and E [n 0(x 0)2]<∞.

In this framework the metric coe?cient M (ξ)is not changed with respect to 1+n (x )to leading order in ε,but there is a correction of order ε2described in the following lemma.Lemma 3.1.The metric coe?cient M (ξ)can be expanded as (3.3)M (ξ)=1+εn 0(ξ)+ε2(n 1(ξ)?γ0(0))+o (ε2),where n 1is a zero-mean random process and γ0(x )=E [n 0(x 0)n 0(x 0+x )]is the autocorrelation function of n 0.Proof .The process M (ξ)is de?ned by (2.3).It is given by the convolution of a

smooth sech 2kernel with the composition of the random process n 0and the real part of the conformal map x (ξ,ζ)evaluated at the unperturbed bottom ?√β=?√β0ε.The imaginary part y (ξ,ζ)satis?es the Laplace equation Δy =0in the domain R ×(?√β0ε,0)and the Dirichlet boundary conditions y (ξ,0)=0and y (ξ,?ε√β0)=?ε√β0[1+εn 0(x (ξ,?ε√β0))].Of course,at zeroth order,we have x =ξand y =ζ.By Fourier transform we can ?nd the following representation for y :y (ξ,ζ)=ζ+ε2√β02π sinh(kζ)sinh(kε√β0)?n 1(k )e ikξdk,where ?n 1(k )is the Fourier transform (in ξ)of n 0(x (ξ,?√β0ε).Using the Cauchy–Riemann equation x ξ=y ζgives x ξ(ξ,ζ)=1+ε2√β02π k cosh(kζ)sinh(kε√β0)?n 1(k )e ikξdk.Since ζis of order ε,we can expand the cosh and the sinh to obtain x ξ(ξ,ζ)=1+εn 0(x (ξ,?√β0ε))+o (ε2)for all ζ∈[?ε√β0,0].Note that if we assume that n 0is

twice di?erentiable,then the o (ε2)is O (ε3).Since M (ξ)=x ξ(ξ,0)by de?nition [30],

this gives the ?rst corrective term of the expansion (3.3).We can also obtain a precise description of the higher-order correction.The process X εde?ned by X ε(ξ):=x ξε2,? β0ε ?ξε2

1002J.GARNIER,J.C.MU?NOZ GRAJALES,AND A.NACHBIN satis?es the random ordinary di?erential equation(ODE)

dXεdξ=

ε2

+Xε

+o(1).

By applying a di?usion-approximation theorem[32,33],Xε(ξ)converges in distribu-tion asε→0to the di?usion process X(ξ)with the in?nitesimal generator

L= ∞

n0(x)

n0(u+x)

du=

∞

γ0(u)du

?x2

∞

γ 0(u)du

that can be identi?ed as

X(ξ)=

2μ0Wξ?γ0(0)ξ,

whereμ0= ∞

γ0(u)du and Wξis a standard Brownian motion.This shows that the

drift of X is constant and equal to?γ0(0),which gives the

result.

In the forthcoming asymptotic analysis,the most important term in the expansion (3.3)isεn0(ξ),since it is the one that is responsible for the coupling between left-and right-going modes.The contribution of the zero-mean process n1vanishes in the limitε→0.The correction to the average depth

E[M(ξ)]=1?ε2E[n0(0)2]+o(ε2)

will play a role,because it induces a change in the average velocity of orderε2,which gives a shift of order1after a propagation distance of orderε?2.

Our goal is to study the wave propagation for times and distances of orderε?2. Accordingly in(2.9)–(2.10)we can neglect the terms of orderε3.We can also use (2.14),valid up to orderε,to rewrite someξderivatives as time derivatives.This can be done with a su?cient accuracy for the nonlinear and dispersive terms.As a result,

we obtain

?ξ

=Q(ξ)

Mξ

1?1

?11

+ε2

α0

3A+B0

0A+3B

+ε2

β0

?10

+O(ε3),

(3.4)

where Mξis a zero-mean process of orderεand Q is given by(2.15).The?rst step of the derivation of the e?ective equation for the front pulse is based on a series of transformations to rewrite the evolution equations(3.4)of the modes by centering along the characteristic of the right-going mode.This gives an upper-triangular sys-tem that can be integrated more easily.In a second step an averaging theorem[4,18] is applied to this system to establish an e?ective nonlinear equation for the front pulse for times and distances of orderε?2.These computations follow the lines of the proof of the front pulse analysis in the random,nonlinear,hyperbolic case given in[12]. The result can be stated as follows.

Proposition3.2.Let?B0be the solution of the deterministic equation

(3.5)??B0

?ξ

=L?B0+

3α0

??B0

?τ

β0

?3?B0

?τ3

SOLITARY WAVES OVER RANDOM TOPOGRAPHY1003 starting from?B0(0,τ)=f(τ).The front pulse Bε(ξ,τ):=B(ξ/ε2,τ+ξ/ε2),ξ∈[0,∞),τ∈R,converges in distribution in the space of the continuous functions (equipped with the topology associated with the supremum norm over the compact intervals)to?B given by

(3.6)?B(ξ,τ)=?B0(ξ,τ?Tξ),

where Tξis the random time shift

(3.7)Tξ=

√

Wξ+

γ0(0)

ξ.

Here Wξis a standard Brownian motion,and the operator L can be written explicitly in the Fourier domain as

∞

?∞L B(τ)e iωτdτ=?b0(2ω)ω

∞

?∞

B(τ)e iωτdτ,

(3.8)

b0(ω)=

∞

0γ0(x)e iωx dx,

(3.9)

whereγ0(x)=E[n0(x0)n0(x0+x)]is the autocorrelation function of n0.

The random time shift Tξoriginates from the random propagation speed.L is a pseudodi?erential operator that models the deterministic pulse deformation.It can be interpreted as an e?ective pseudoviscosity originating from the random forcing.The e?ective equation for the front pulse depends on randomness(through the function b0),on dispersion(throughβ0),and on nonlinearity(throughα0).

First,the pseudodi?erential operator L satis?es a special but rather intuitive time property.Indeed,in the time domain,we can write

L B(τ)=

γ0

1[0,∞)(τ)

?2B

?τ2

(τ)

∞

γ0

?2B

?τ2

(τ?s)ds.

The indicator function1[0,∞)is essential to interpret the convolution correctly.It means that the e?ective viscosity cannot di?use the wave energy in the forward direc-tion(ahead the front)but only in the backward direction(behind the front).This in turn implies that the reduction of the pseudodi?erential operator L to a second-order di?usion operator that we discuss next should be handled with precaution.

We now discuss further properties of the pseudodi?erential operator L.It can be pided into two parts L=L r+L i:

∞

?∞L r B(τ)e iωτdτ=?b r(2ω)ω

∞

?∞

B(τ)e iωτdτ,

(3.10)

∞

?∞L i B(τ)e iωτdτ=?ib i(2ω)ω

∞

?∞

B(τ)e iωτdτ,

(3.11)

where b r and b i are,respectively,the real and imaginary part of b0:

b r(ω)=

∞

0E[n0(0)n0(x)]cos(ωx)dx,b i(ω)=

∞

E[n0(0)n0(x)]sin(ωx)dx.

By the Wiener–Khintchine theorem[24],b r is proportional to the power spectral density of the random stationary process n0.As a result,b r is nonnegative,which

Copyright ? by SIAM. Unauthorized reproduction of this article is prohibited. 1004J.GARNIER,J.C.MU ?NOZ GRAJALES,AND A.NACHBIN shows that L r can be interpreted as an e?ective di?usion operator.More precisely,for small frequencies,L r behaves like a second-order di?usion.Indeed,if ωl c 1,then b r (ω) μ0,where μ0:= ∞0γ0(x )dx ,and L r μ04?2?τ2.On the other hand,b r decays to zero for high frequencies ωsuch that ωl c 1.Indeed,we have assumed that E [n 20(0)]<∞,which is equal to ?γ 0(0).By use of the inverse Fourier transform this shows that ω2b r (ω)dω<∞and ω2b r (ω)should decay fast enough as ωgoes to in?nity to ensure the convergence of this integral.As a result L r

has no e?ect on the high-frequency components.L i is an e?ective dispersion operator,since it preserves the energy.It behaves like a third-order dispersion for small frequencies.Indeed,if ωl c 1,then b i (ω) ωβ1,where β1:= ∞0xγ0(x )dx ,and L i ?β12?3?τ.Furthermore,similarly as for b r ,b i decays to zero for high frequencies,so that L i has no e?ect on the high-frequency components.It is interesting to determine which operator,L r or L i ,is the more important one.

By scaling arguments,we get that ω3β1is of the order of (ωl c )μ0l 2c ,which is smaller

than μ0ω2if ωl c 1.As a result,the e?ective dispersion for small frequencies is usually smaller than the e?ective di?usion.Furthermore,we usually have β1>0.This is the case,for instance,for a Gaussian autocorrelation function of the form γ0(x )=exp ?πx 2/l 2c .The length l c is indeed the correlation length of the medium in the sense of (2.5).Besides,we have μ0=l c /2,and

β1=l 2c /(2π).The fact that β1>0shows that the dispersion is reduced compared to the original one:the third-order dispersion coe?cient,which is equal to β0/6in the absence of randomness,takes the value β0/6?β1/2in the presence of random topography.This dispersion reduction will be illustrated in the numerical simulations reported at the end of the paper.Note,however,that special con?gurations can be encountered that do not belong to the general case described above.One interesting case deserves a speci?c discussion.Let us consider for a while that the process n 0is the derivative of a smooth stationary zero-mean random process νsuch as a Gaussian random process

with Gaussian autocorrelation function.We then have γ0(u )=??2u E [ν(0)ν(u )],and μ0=0while β1=?E [ν(0)2]<0.This shows that,in this very particular case,the dominant operator is the dispersion operator,and it enhances the original dispersion.Our approach allows us to get the complete statistical distribution of the free surface wave,which gives more precise results than the mean-?eld approach.Indeed,

by Proposition 3.2,the mean ?eld B mf (ξ,τ)=lim ε→0E B (ξ/ε2,τ+ξ/ε2) satis?es

(3.12)?B mf ?ξ=L mf B mf +3α04B mf ?B mf ?τ+β06?3B mf ?τ3,where (3.13)L mf =L i +L r +b 0(0)4?2?τ2

.The additional di?usion originates from the averaging with respect to the random

time delay T ξde?ned by (3.7).In [23]the authors obtained (3.12)–(3.13)by applying

SOLITARY WAVES OVER RANDOM TOPOGRAPHY1005 a multiscale expansion technique to the mean?eld.Our results clearly show that di?usion is overestimated by the mean-?eld approach and that only L r is a physical di?usion.

To sum up,in the general case whereμ0>0,if the typical wavelength of the original pulse f is larger than the correlation radius of the medium,then the early steps of the e?ective evolution equation are those of the viscous KdV equation

(3.14)??B0

?ξ

μ0

?2?B0

?τ2

3α0

??B0

?τ

β0

?β1

?3?B0

?τ3

In this case we have an eddy viscosity[34]which looks like a kinematic viscosity.

Both forms of viscosity remove energy from the coherent wavefront in a di?usive-like

manner.We can think of a low-pass Gaussian?lter.However,the energy?ltered by

a kinematic viscosity is lost forever and cannot be recovered:it models an irreversible

process.As discussed in a recent letter[15]the energy?ltered by the eddy viscosity

corresponds to a conversion of coherent energy transported by the front pulse into

incoherent energy contained in the small wave?uctuations following the front pulse.

The energy?ltered by the eddy viscosity can be recovered along the coherent wavefront

by a time-reversal recompression using a time-reversal mirror.This surprising result

shows that the eddy viscosity models a reversible process.Therefore,although the

kinematic and eddy viscosity appear with the same form,they are of a very di?erent

nature.

Finally,note that(3.14)may be valid only during the early steps of the wave

propagation.Indeed,new wavelengths generated by the nonlinearity may lie in the

tail of the function b0(2ω),and it is then necessary to take into account the complete

pseudodi?erential operator,not only its expansion.Note also that we cannot consider

the true white noise case,because a white noise does not ful?ll the boundedness

requirement that is necessary to ensure the convergence result.Indeed,the assumption E[n0(0)2]<∞(and E[n 0(0)2]<∞)is important for the proof of the convergence result,and it is not ful?lled by the white noise whose variance is in?nity.Nevertheless,

qualitatively speaking,white noise disorder would a?ect the entire spectrum of the

pulse as opposed to the case discussed here.

3.2.Rough topography.We now consider a speci?c case of rough topography

to emphasize that the class of topographies that can be addressed by our theory is

broader than those considered in the previous section.It is well known that stepped

pro?les,joining regions of di?erent depths,is a di?cult problem of interest,since

this corresponds to waves propagating over,for instance,the continental shelf[9,10].

Also many times stepped topographies are considered as an approximation strategy

for arbitrary bottom slopes[10,7].Thus we will present a case of a randomly stepped

pro?le for which certain statistical quantities can be computed explicitly.Moreover,

as in the linear problems considered in[10,7]we consider the wide-spacing hypothesis

in the sense that the jumps are separated by a distance of order one.

Namely,in this section we assume that the topography is a rough,stepped pro?le,

in the sense that the random process n0is bounded and satis?es E[n0(0)2]<∞,but

it has jumps.For the sake of simplicity,we shall assume that n0can be modeled by (3.15)n0(x)=n∞(?1)N x,

where(N x)x≥0is a Poisson process with intensity1/l c.This means that(i)N x

takes integer values;(ii)the increments of the process are independent;and(iii)the

1006J.GARNIER,J.C.MU?NOZ GRAJALES,AND A.NACHBIN

distribution of N x is P(N x=k)=exp(?x/l c)(x/l c)k k!,k∈N.As a result,the process n0is Markov,takes values in{?n∞,n∞},and has mean zero and variance n2∞,and its autocorrelation function is

E[n0(x0)n0(x0+x)]=n2∞exp

?2|x|

l c

Note that l c is indeed the correlation length of the medium in the sense of(2.5). Contrarily to the smooth topography case studied in the previous section,the metric process M(ξ)here is di?erent from1+εn0because the jumps of n0are smoothed. The analysis of the smoothing can be carried out by considering?rst the case where

n0,1(x)=n∞sgn(x),

which is an elementary jump from?n∞to n∞.Using the Schwarz–Christo?el for-mula[30]we get the following representation for the corresponding metric coe?cient:

M1(ξ)=C ?

?cosh(

πξ

√

β0ε

)

cosh(π(ξ?a)

√

β0ε

)

1/2

where

1?ε2n2∞,a=

√

β0ε

1+εn∞

1?εn∞

An expansion with respect toεgives to leading order

M1(ξ)=1+εn∞tanh

πξ

√

β0ε

which can also be written in the convolution form

(3.16)M1(ξ)=1+εKε?n0,1(ξ),Kε(ξ)=

√

β0εcosh2

πξ

√

β0ε

Since the random process n0de?ned by(3.15)is stepwise constant and its jumps are separated by a distance of order one,the previous analysis gives the leading-order expression for the metric coe?cient M:

(3.17)M(ξ)=1+εKε?n0(ξ),

where Kεis the kernel(3.16).The derivative of M has the form

M (ξ)=πn∞

√

β0

cosh2

πξ

√

β0ε

∞

j=1

(?1)j

cosh2

π(ξ?X j)

√

β0ε

where the X j are the random positions of the jumps of n0.From a statistical point of view,X0=0and(X j?X j?1)j≥1is a sequence of independent and identically distributed random variables with exponential distribution and mean l c.The deriva-tive M (ξ)is therefore a collection of alternating negative and positive peaks with the same sech2shape,amplitude of order1,and width of orderε.Although the scaling

SOLITARY WAVES OVER RANDOM TOPOGRAPHY1007 regime is di?erent,the same analysis as in the previous section can be carried out, leading to the same result as stated in Proposition3.2.The real and imaginary parts of the Fourier transform of the pseudodi?erential operator L here are given by

?b r(2ω)ω2

4=?

n2∞l cω2

8(1+ω2l2c)

b i(2ω)ω2

n2∞l2cω3

8(1+ω2l2c)

Note that the di?usive and dispersive parts of the pseudodi?erential operator do not vanish whenωl c 1,but they behave like

?b r(2ω)ω2

4→?n

∞

8l c

b i(2ω)ω2

→?n

∞ω

As a consequence,when the correlation length l c of the random topography is larger than the pulse width,then the e?ective pulse front equation has the form of a KdV equation with damping:

(3.18)??B0

?ξ

∞

??B0

?τ

n2∞

8l c

3α0

??B0

?τ

β0

?3?B0

?τ

The analysis of the random topography of the form(3.15)shows that the regulariza-tion e?ect due to stochastic forcing is more important for a rough topography than for a smooth one.This analysis is also interesting because it allows an explicit char-acterization of the regularization e?ect of the pseudodamping.Indeed,in the special caseβ0=0,a closed form expression for the solution of(3.18)can be obtained by the method of characteristics[36]

??B0?τ(ξ,τ)=

exp(?cξ

l c

)f (τ+cξ)

1?3α0l c

[1?exp(?cξ

l c

)]f (τ+cξ)

where f is the initial pulse shape(supposed to be smooth)atξ=0,f is its derivative, and c=n2∞/8.This expression of the(derivative of the)solution is valid as long as the denominator does not vanish.As a consequence,the shock formation is determined by the value of the parameter

S:=6α0l c

n2∞

max

τ∈R

f (τ).

(1)If S>1,then a shock occurs at the propagation distance

ξs=8l c

n∞

S?1

(2)If S≤1,then the solution of(3.18)is global.The e?ective damping removes energy fast enough to prevent from the shock formation.

Note that the caseβ0=0is a limit case that is never reached in practice,so that shock formation can never be observed because of the presence of dispersion. However,it is a limit case that is mathematically well known and investigated,and it can be treated analytically.Besides,the parameterβ0can be very small,and then it is interesting to study how damping competes with nonlinearity.

1008J.GARNIER,J.C.MU?NOZ GRAJALES,AND A.NACHBIN

815cedc0d5bbfd0a79567398rge-amplitude,rapidly varying topography.

4.1.Statement of the main results.In this section we still write the small nonlinear and dispersion parameters asα=ε2α0andβ=ε2β0,but we assume that the random topography has a correlation length smaller than the wavelength.The suitable scaling that will give rise to a nontrivial asymptotic regime consists in taking

(4.1)n(x)=εp n0 x

where n0is a stationary random process with standard deviation and correlation length of order1,and p∈(3/8,1/2).Equation(4.1)models a rapidly varying random topography.

1.The assumption p<1/2ensures that the correlation lengthε2?2p of the

random bottom is much smaller than the scaleεof the smoothing kernel in

(2.3).The smoothing e?ect then plays a primary role.Note that if p>1/2,

then the correlation length is much larger than the scale of the smoothing kernel,and the smoothing e?ect is negligible.The metric coe?cient is of the form M(ξ)=1+εp m(ξ/ε2?2p)with m n0to leading order.We then get the same results as in section3.

2.The restriction p>3/8allows us to obtain a universal statistical description

of the metric coe?cient M,in the sense that it does not depend on the detail of the statistics of n0.When going to p<3/8,the statistics become more complicated,as we shall see below.

3.The amplitude factorεp is chosen so that the integrated covariance of the

random process n is of orderε2,i.e.,of the same order as the dispersion and nonlinearity parameters.The e?ects of the random perturbations then become of order one after a propagation distance of orderε?2.

4.In terms of smoothness we assume that there exists a constant C∈(0,1)such

that n0 ∞≤C almost surely,and we require n0to be di?erentiable with E[n 0(0)2]<∞.We also assume that the random process possesses strong mixing properties,more precisely,that it isφ-mixing withφ∈L1/2(see[19, pp.82–83]).

In this framework,the scale of variations of the metric coe?cient M(ξ)is di?erent from the original scale of the topographic coe?cient n,as shown by the following proposition.This new scale is imposed by the convolution of the fast topographic process with the scaled sech2function.

Proposition 4.1.The orography-dependent coe?cient M(ξ)in the case of a large-amplitude rapidly varying random topography(4.1)acquires Gaussian statistics in the limitε→0,with the asymptotic autocorrelation function

(4.2)1

E[(M(ξ0)?1)(M(ξ0+εξ)?1)]ε→0

?→γm(ξ),

where

(4.3)γm(x)=πμ0

√

β0

πx

√

β0

cosh(πx

√

β0

)?sinh(πx

√

β0

)

sinh3(πx

√

β0

)

andμ0is the integrated correlation function of the random topography:

μ0=

∞

E[n0(0)n0(x)]dx.

SOLITARY WAVES OVER RANDOM TOPOGRAPHY1009 Sections4.2–4.3are devoted to the proof of Proposition4.1.Note thatμ0is the only parameter that remains from the random topography.This result shows that the wave dynamics acquires a universal behavior driven by a random Gaussian process with the autocorrelation function(4.3).The power spectral density of this process, given by the Fourier transform of the autocorrelation function,is

?γ0(k)=μ0 √

β0k

sinh(

√

β0k)

As a by-product of the analysis carried out in the proof of Proposition4.1,we obtain the expansion of the averaged value of the metric coe?cient M(ξ)in the limit ε→0.

Proposition4.2.The averaged value of the metric coe?cient can be expanded

asε→0:

(4.4)E[M(ξ)]=1?ε4p?1√

β0

2π

?γ0(k)|k|dk+o(ε4p?1),

where?γ0is given by(4.9).

This result can be used to prove a numerical observation reported in the literature and also to comment on the range of validity of Proposition4.1.Note that the coe?cient?γ0is the Fourier transform of the autocorrelation function of the process n0,which is nonnegative by the Wiener–Khintchine theorem.This shows that the ?rst-order correction to the expectation of the metric coe?cient is negative valued, which means that the e?ective bottom is located at a depth<1.The fact that a zero-mean varying random topography can give rise to a nonzero average depth through the conformal mapping was already pointed out in[28].In our setting p∈(3/8,1/2),this e?ect is negligible.However,as p→3/8,the expansion(4.4)shows that the average value of M?1becomes of orderε1/2,which is of the same order as the standard deviation of the asymptotic zero-mean Gaussian process described in Proposition4.1. This remark shows that the situation becomes more complicated when p→3/8,and it will be addressed in a future work.

Finally,we go back to the wave propagation problem using the results above.If we focus our attention on the coherent front,then the picture becomes rather simple and qualitatively similar to the one encountered in the case of a small-amplitude slowly varying topography addressed in section3.To summarize,up to a random time shift, the dynamics of the coherent front pulse is governed by a deterministic di?usive KdV-type equation whose di?usion coe?cient depends only on the randomness through the parameterμ0.

Proposition 4.3.The front pulse Bε(ξ,τ):=B(ξ/ε2,τ+ξ/ε2)converges to ?B(ξ,τ)=?B

(ξ,τ?Tξ),where Tξis the random time shift de?ned by(3.7)and?B0is the solution of the deterministic equation

(4.5)??B0

?ξ

μ0

?2?B0

?τ2

3α0

??B0

?τ

β0

?3?B0

?τ3

starting from?B0(0,τ)=f(τ).

The proof of Proposition4.3is given in subsection4.4in the linear case.Its nonlinear extension is then suggested by a heuristic argument.Note that the variance of the random time delay as well as the e?ective noise-induced di?usion depend only on the parameterμ0,which is the integrated autocorrelation function of the process n0.

Copyright ? by SIAM. Unauthorized reproduction of this article is prohibited. 1010J.GARNIER,J.C.MU ?NOZ GRAJALES,AND A.NACHBIN 4.2.Asymptotic analysis of the conformal mapping.The metric coe?-

cient M (ξ)is de?ned by (2.3).It is given by the convolution of a smooth explicit kernel with the composition of the random process n 0and the real part of the confor-mal map x (ξ,ζ)evaluated at the unperturbed bottom ?√β.The goal of this section is to get an asymptotic expansion of x (ξ,?√β)as ε→0.This requires us to analyze the conformal mapping introduced in [29,30].We start by considering the problem

(4.6)Δξ(x,y )=0,with the boundary condition ξy =0at y =0and (4.7)ξn =ξy + βn (x )ξx =0at y =? β(1+n (x )).To solve this problem in the asymptotic ε→0,we ?rst replace the boundary condition at the random bottom y =?√β(1+n (x ))by a boundary condition at the ?at bottom y =?√β=?ε√β0:ξy + β0ε3p ?1n 0 x ξx =R ε1+R ε2,with R ε1= β0ε1+p n 0 x ε2?2p 10ξyy x,? β0ε 1+θεp n 0 x ε2?2p dθ,R ε2=β0ε4p n 0n 0 x ε2?2p 10ξxy x,? β0ε 1+θεp n 0 x ε2?2p dθ.The corrective terms R ε1and R ε2are the Lagrange remainders of the Taylor expansions of ξy and √β0ε3p ?1n 0(x ε2?2p )ξx at y =?ε√β0.The solution has the form (4.8)ξ(x,y )=ξ0(x )+ξ1(x,y )+ξr (x,y ),

where ξ0(x )=x is the identity describing the conformal map in the absence of per-

turbation,and ξ1is the ?rst-order corrective term satisfying the Laplace equation Δξ1=0,the boundary condition ξ1y =0at y =0,and the boundary condition ξ1y +√β0ε3p ?1n 0(x/ε2?2p )ξ0x =0at y =?√β0ε.After a Fourier transform with respect to x ,we ?nd that ξ1(x,y )=i √β0ε1+p 2π cosh(kyε2p ?2)sinh(k √β0ε2p ?1)e ik x ε2?2p ?n 0(k )dk ,where the Fourier transform is de?ned by ?n 0(k )= n 0(x )e ?ikx dx.At the unperturbed bottom y =?√β0εthe random correction ξ1is a zero-mean

process (in x )with autocorrelation function E ξ1(x 0,?ε 0ξ1(x 0+ε2?2p x,?ε 0 =β0ε2+2p 2π ?γ0(k )e ikx dk +o (ε2+2p )=β0ε2+2

p γ0(x )+o (ε2+2p ),

SOLITARY WAVES OVER RANDOM TOPOGRAPHY1011 where

(4.9)γ0(x)=E[n0(x0)n0(x0+x)]

is the autocorrelation function of the process n0,and?γ0is its Fourier transform,which is proportional to the power spectral density of the random process n0:

?n0(k)?n0(k+k )

=2π?γ0(k)δ(k ).

This result also shows that Rε1=O(ε6p?2)and Rε2=O(ε9p?3),which means that the terms Rεj are higher-order corrections in(4.7)in the case p>1/3.It also shows that ξr=O(ε4p).Inverting relation(4.8)at the undisturbed bottom,we have

(4.10)x(ξ,?ε

=ξ+ε1+p Xε(ξ)+O(ε4p),

with Xε(ξ)a fast varying(at scaleε2?2p)zero-mean process with standard deviation of order one given by

(4.11)Xε(ξ)=?i

√

β0

2π

cosh(k

√

β0ε2p?1)

sinh(k

√

β0ε2p?1)

e ikξ

ε2?2p?n0(k)dk.

This expression will be used in the next section to study the asymptotic properties of the metric coe?cient.

4.3.Asymptotic analysis of the metric coe?cient.In this section we prove Propositions4.1and4.2.First note that,from(2.3)and(4.10),the expected value of the metric coe?cient M can be expanded as

E[M(ξ)]=1+πε4p?2

4√

β0

E[n 0(ξ0ε2p?2)Xε(ξ0)]

cosh2

√

β0ε

(ξ?ξ0)

dξ0+···,

where Xεis de?ned by(4.11).In the asymptoticε→0,we obtain(4.4),which proves Proposition4.2.

In order to prove Proposition4.1,we are going to show that,for anyξ and for any smooth test function g,the sequence of random variables

(4.12)Gεξ (g)=

√

(M(ξ +εξ)?1)

g(ξ)dξ

converges in distribution asε→0to a Gaussian random variable whose variance can be identi?ed.This convergence result will be obtained by computing the limiting

moments of Gε

ξ (g).

From the de?nition(2.3)of the coe?cient M,the complete expression of Gε

ξ (g)

(4.13)Gεξ (g)=

√

β0ε3/2?p

n0(x(ξ0)

ε2?2p

)g(ξ)

cosh2

√

β0ε

(ξ0?ξ ?εξ)

dξ0dξ.

By(4.10),we?rst note that

x(ξ0)ε2?2p =

ξ0

ε2?2p

+O(ε3p?1).

1012J.GARNIER,J.C.MU?NOZ GRAJALES,AND A.NACHBIN

Since n 0can be bounded(in quadratic mean),it is possible to substitute n0(ξ0

ε2?2p

)

for n0(x(ξ0)

ε2?2p )in(4.13)up to an error of orderεp?3/2×ε3p?1×ε=ε4p?3/2.Since

p>3/8,this error is negligible,and we get that,to leading order inε,

Gεξ (g)=

√

β0ε3/2?p

n0(ξ0

ε2?2p

)g(ξ)

cosh2

√

β0ε

(ξ0?ξ ?εξ)

dξ0dξ.

We next perform the change of variableξ0→u0=(ξ0?ξ )/ε2?2p:

Gεξ (g)=πε1/2?p

√

β0

n0(ξ

ε2?2p

+u0)g(ξ)

cosh2

√

β0

(ε1?2p u0?ξ)

du0dξ.

We denoteε =ε1?2p.Remember that p<1/2so thatε goes to zero asεgoes to zero.We?x q∈N and consider the2q th 815cedc0d5bbfd0a79567398ing the stationarity of the process n0,we have

Gεξ (g)2q

π2ε

16β0

ξ1,...,ξ2q

u1,...,u2q

j=1

n0(u j)]

g(ξj)

cosh2

√

β0

(ε u j?ξj)

du j dξj.

The integrand in this equation is symmetric with respect to the set(u i)i=1,...,2q,and so we can write

Gεξ (g)2q

π2ε

16β0

(2q)!

ξ1,...,ξ2q

u1<···

j=1

n0(u j)]

g(ξj)

cosh2

√

β0

(ε u j?ξj)

du j dξj.

By setting u2i?1=v2i?1/ε and u2i=u2i?1+v2i,

Gεξ (g)2q

π2

16β0

(2q)!

ξ1,...,ξ2q

i=1

n0(v2i?1

)n0(v2i?1

+v2i)

i=1

g(ξ2i?1)g(ξ2i)

i=1

cosh2

√

β0

(v2i?1?ξ2i?1)

cosh2

√

β0

(v2i?1+ε v2i?ξ2i)

i=1

dv i dξi.

We now use the strong mixing properties of the process n0,which implies that,for any v1

i=1

2i?1

+v2i

ε→0

?→

i=1

γ0(v2i),

whereγ0(x)=E[n0(x0+x)n0(x0)].As a result,

Gεξ (g)2q

ε→0

?→

π2

16β0

(2q)!

ξ1,...,ξ2q

v2>0,...,v2q>0

i=1

γ0(v2i)g(ξ2i?1)g(ξ2i)

cosh2

√

β0

(v2i?1?ξ2i?1)

cosh2

√

β0

(v2i?1?ξ2i)

dv2i?1dv2i dξ2i?1dξ2i.

Copyright ? by SIAM. Unauthorized reproduction of this article is prohibited. SOLITARY WAVES OVER RANDOM TOPOGRAPHY 1013The multiple integral is symmetric with respect to the set (v 2i ?1)i =1,...,q ,and so we

can write E G εξ (g )2q ε→0?→ π216β0 q (2q )!q ! ξ1,...,ξ2q v 1,v 3,...,v 2q ?1 v 2>0,...,v 2q >0×q i =1γ0(v 2i )g (ξ2i ?1)g (ξ2i )cosh 2 π2√β0(v 2i ?1?ξ2i ?1) cosh 2 π2√β0(v 2i ?1?ξ2i ) dv 2i ?1dv 2i dξ2i ?1dξ2i .This multiple integral can now be factorized:E G εξ (g )2q ε→0?→ π216β0 q (2q )!q !×?? v 1 v 2>0 ξ1,ξ2γ0(v 2)g (ξ1)g (ξ2)cosh 2 π2√β0(v 1?ξ1) cosh 2 π2√β0(v 1?ξ2) dv 1dv 2dξ1dξ2??q .By integrating the integral with respect to v 1and v 2,we get (4.14)E G εξ (g )2q ε→0?→(2q )!2q !× ξ1,ξ2γm (ξ1?ξ2)g (ξ1)g (ξ2)dξ1dξ2 q ,where γm is given by (4.3).Let us now consider an odd moment of the form E [G εξ (g )2q +1],where q is a nonnegative integer.This moment involves the expectation of the product of an odd number of terms n 0(x (ξ )/ε +u i ),and following the same lines as the previous analysis,we get that this expectation converges to 0as ε→0,so that we obtain

(4.15)E G εξ (g )2q +1 ε→0?→0.The limits (4.14)–(4.15)are the moments of a zero-mean Gaussian random variable with variance ξ1,ξ2γm (ξ1?ξ2)g (ξ1)g (ξ2)dξ1dξ2;thus G εξ (g )converges in distribution to this random variable.This holds true for any test function g .As a result,this shows that (ε?1/2(M (ξ +εξ)?1))?∞<ξ<∞converges to a Gaussian process.The

limiting Gaussian process has zero mean and its autocorrelation function is γm .4.4.Asymptotic analysis of the front pulse.As shown in the previous sec-tion,in the regime where the random topographic coe?cient n (x )has the form (4.1)with p ∈(3/8,1/2),the metric coe?cient M is asymptotically Gaussian distributed with the autocorrelation function (4.2)–(4.3).Accordingly we can write (4.16)M (ξ)=1+ε1/2m ξ ,where m is a zero-mean smooth stationary random process with the autocorrelation function γm .The variance of m is E [m (0)2]=γm (0)=πμ0/(3√β0).We also know that m is Gaussian distributed,but this hypothesis is not used in the derivation of the results of this section.The expression (4.16)is the leading-order term of the random coe?cient M (ξ).As is usual in the asymptotic analysis of randomly forced ODEs,the higher-order zero-mean corrections play no role and vanish in the limit ε→0.

The deterministic corrections like (4.4)survive in the limit ε→0,and here they take

1014J.GARNIER,J.C.MU?NOZ GRAJALES,AND A.NACHBIN

the form of a deterministic shift.This can be explained by the fact that the average bottom,and therefore the average velocity,are modi?ed.We will not discuss this point further,as we are mainly interested in the wave deformation in this paper.

We now revisit the derivation of the e?ective equation for the front pulse in the case where the metric coe?cient has the form(4.16).We?rst expand the system by keeping the leading-order terms inε(i.e.,those which play a role for propagation distances of the order ofε?2):

??ξ

=Q(ξ)

2ε1/2

1?1

?11

+ε2

α0

3A+B0

0A+3B

+ε2

β0

?10

?t3

+ε1/2

β0

?11

(4.17)

where

(4.18)Q(ξ)=

1+ε1/2

m(ξ

)ε1/2

m(ξ

)

?ε1/2

m(ξ

)?1?ε1/2

m(ξ

)

The third and fourth terms(nonlinear and dispersive)of the right-hand side of(4.17) are similar to the ones appearing in(3.4),but the other terms are rather di?erent. In particular the second term involves highly?uctuating random components,with amplitudes of orderε?1/2,and the asymptotic analysis requires more elaborate tools than in the slowly varying case addressed in the previous section,where the amplitude of the random?uctuations was of orderε.An e?cient and rigorous study can be performed in absence of nonlinearity,and we present this study below.

We assume for a while thatα0=0.We apply a Fourier transform with respect to time,

(4.19)?A(ξ,ω)=

A(ξ,t)e iωt dt,?B(ξ,ω)=

B(ξ,t)e iωt dt,

so that the PDE(4.17)can be reduced to an in?nite set of ODEs for the modes ?A and?B:

??ξ ?

iωQ(ξ)+

√

1?1

?11

?iω3ε2β0

?10

+iωε1/2

β0

?11

(4.20)

The boundary conditions corresponding to a right-going pulse coming from the ho-mogeneous half-spaceξ<0are

?B(0,ω)=?f(ω),?A(L/ε2,ω)=0.

The problem,from the analytic point of view,is very simple because each frequency-dependent pair of ODEs is uncoupled(from the rest)and linear.However,from the statistical point of view,these ODEs are coupled because they share the same process m(and its derivatives).The analysis of the correlation between di?erent frequency components plays a central role in the convergence result.

Copyright ? by SIAM. Unauthorized reproduction of this article is prohibited. SOLITARY WAVES OVER RANDOM TOPOGRAPHY 1015We next rescale the propagation distances and introduce the left-and right-going modes centered along the deterministic characteristic lines:

?A ε1(ξ,ω)=?A ξε2,ω exp ?iωξε2 ,?B ε1(ξ,ω)=?B ξε2,ω exp iωξε2 ,so that the system now has the form ??ξ ?A ε1?B ε1 = 12ε5/2m ξε3 1?e ?2iωξε2?e 2iωξε21 +iω2ε3/2m ξε3 1e ?2iωξε2?e 2iωξ2?1 ?iω3β06 ?1001 +iωβ012ε3/2m ξε3/2 ?1e ?2iωξε2?e 2iωξε2

1 ?A ε1?B ε1 .(4.21)The ?rst term of the right-hand side,of the form ε?5/2q (ξ/ε3),is not written in a

form suitable for the application of a di?usion-approximation theorem,which requires a scaled random term of the form ε?p/2q (ξ/εp ).It is possible to rewrite the ?rst term in a more convenient form.Noting that 1ε5/2m ξε3 e ?2iωξε2=d dξ ε1/2m ξε3 e ?2iωξε2 +2iωε3/2m ξε3 e ?2iωξε2,1εm ξε e 2iωξε2=d dξ ε1/2m ξε e 2iωξε2 ?2iωεm ξε e 2iωξε2,we can rewrite the system as (4.22)??ξ ?A ε1?B ε1 =[M ε(ξ)+N ε(ξ)] ?A ε1?B ε1 ?iω3β06 ?1001 ?A ε1?B ε1 ,where M ε(ξ)= M ε1(ξ)M ε2(ξ)M ε2(ξ)M ε1(ξ) ,N ε(ξ)= N ε1(ξ)N ε2(ξ)N ε2(ξ)N ε1(ξ) ,M ε1(ξ)=iω2ε3/2 m ξε3 ?β06m ξε3 ,N ε1(ξ)=ε1/22d dξ m ξε3 ,M ε2(ξ)=?iω m ξ ?β0m ξ e ?2iωξε2,N ε2(ξ)=?ε1/2d m ξ e ?2iωξε2 .Note that the scaling of the matrix M εis appropriate for the application of an di?usion-approximation theorem,while the matrix N εis the derivative of a process of order √ε.The quantity that we are interested in is the front pulse:(4.23)B L ε2,L ε2+t =12π ?B 1 L ε2,ω e ?iωt dω.We thus recover the situation studied in [6],which dealt with the hyperbolic acoustic

equations.However,there are three di?erences:

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