Evolution Solutions of Perturbed Khokhlov-Zabolotskaya Equation

SHI Qihongand CHANG Suqin

Department of Mathematics,Lanzhou University of Technology,Lanzhou 730050,China.

1 Introduction

The canonical equation for weakly nonlinear and weakly diffracting waves is the Khokhlov-Zabolotskaya(KZ)equation[1],which can be written in the general form

whereu=u(x,y,t)is typically a measure of the wave disturbance,xis a spatial variable measured in a frame moving with the wave,yis a transverse spatial variable,andtis a time-like variable.The nconst antαis aquadrati cnonlinearity parameter.It is well known that(1.1)provides an accurate description of the evolution of many systems including those corresponding to acoustic waves in air and water,shallow water waves,acoustic and magnetosonic waves in in nonlinear medium without dispersion or absorption.The study of numerous approximations to the KZ equation in(1.1)has a prominent history concerning the symbiotic interaction of mathematical model and scientific computing to gain insight in the topic.

Actually,for nonlinear optics,Alfv′en waves in magnetohydrodynamics,and shear waves in an isotropic nonlinear solid,the quadratic nonlinearity coefficientαis small or vanishing altogether,(1.1)reduces to

However,if the initial wavefront is curved,near the focal region nonlinear effects will become noticeable.Consequently,Zabolotskaya[2]derived the explicit form of the extension of(1.1)for the case of shear waves propagating in a nonlinear solid in the undisturbed state.Due to the net effect of perturbation analysis,theuuxter min(1.1)is replaced by the cubic term.(1.1)enjoys a new version

which was investigated by Kluwick-Cox[3]and Cramer-Webb[4].Afterwards,a seriesof extensionsystems with weakly relaxing,weakly dissipative,and weakly dispersive were developed,one can refer to[5-9]and the latest result[10].In this paper,we consider a perturbed acoustic wave equation with general nonlinear term and mixed derivative,

wheref(u)=1+un,which is an arbitrary function ofu.γandθare real constants,γdecides the propagation direction onxOyplane,and△⊥denotes the transverse Laplacian in Cartesian coordinates.Our goal is to present a simple and direct method of finding partial exact solutions of KZ type equations with the help of the Hopf equation,which is available to classical KZ equation(1.1).

The rest of this paper is organized as follows.In Section 2,some elementary definitions of Hopf equation have been presented and the implicit solutions of generalized KZ equation are given by mathematical analysis.In Section 3,some examples and relevant numerical simulations are shown at the end of the paper.

2 Outline of the derivation

Here we first recall the solution of Hopf equation.As the special case of Burgers model[11],the general Hopf equation enjoys the following form：

By the hodograph transformationHopf equation can be rewritten towhich implies the solution

whereF(u)is an arbitrary function.

In what follows we return to the KZ equation(1.2).In order to seek for the exact solution,we assume that

where arbitrary numberNof transverses patial coordinateskis an arbitrary non-zero constant.By a computation with respect to the derivatives,

we have

Substituting(2.3)into(1.2),we have

where subscripts denote derivatives with respect to corresponding variables.If we assume that

whereqandmare some constants,then Eq.(2.4)transforms to the equation

The term withqcan be excluded by means of the additional replacement

then

Substituting(2.8)intowe get

then

wheretp,p=1,...,N,are integration constants.Hence Eq.(2.6)transforms to the general Hopf equation

To obtainT(t)in(2.2),substituting(2.7)and(2.9)into△⊥φ=2αn(m-αT/α)gives

Introducingy=αn,we arrive at the Bernoulli equation

which can be solved by a standard method to give

whereCis an integration constant.Ifm=0 then

Therefore,we have reduced findingT(t)to integration of the function in the right side of Eqs.(2.11)or(2.12)whileα(t)=(Tt)1/n.

As a result of the above calculations,the variables in(2.2)can be considered as known and it remains to fi nd the solution of the general Hopf Eq.(2.10).Integration with respect toXgives at once

whereh(T)is an arbitrary fun ctiontobe determined from the initial conditions.Consider the linear differential equation

with the initialU0(X)att=0.Then

whereThe characteristic curve is determined by the equation

Integrating this from 0 toT,we have

Let the initial distributionU(X)atT=0 be given by the functionU0=F-1(X0).Then exclusion ofX0andU0from(2.13)gives the final result：

This equation determines implicitlyuas a function ofx,z,tthrough variables

in terms of two arbitrary functionsF(U)andH(T)which have to be found from the initial conditions.

Remark 2.1.In fact(2.14)is a special solution of Eq.(1.2),because the assumptions in(2.5)lead to the loss of partial solutions.

3 Examples

Example 3.1.We consider the spherical generalized KZ equation

ChooseN=1,k=1,γ=1,q=m=0,t1=t2=0.Then we haveHence Eq.(3.1)has a solution

or

Example 3.2.We consider the generalized KZ equation

ChooseN=2,k=1,γ=1,q=m=0,t1=t2=0.Then we haveHence Eq.(3.3)has a solution

or

We notice that thetpmeans that we consider sound pulse focused in some transverse directions and defocused in the other directions.To the best of our knowledge,the exact solution of the generalized KZ equation have not been consider earlier and we shall apply here our approach of this kind.Therefore,we want to fi nd the solution of Eq.(3.3)propagation of a nonlinear sound pulse that is defocused inzdirection and focused inwdirection.To this end,we take in the above formulasN=2,k=1,q=m=0,t1＜0,t2＞0,h(T)=0.It yields

We also know

where the integration constant is chosen in such a way thatT(0)=0.We assume here that 0≤t≤t2.The self-similar variable has now the form

The variableu(x,y,z,w,t)is expressed in terms ofU(X,T)as

whereUobeys general Hopf equation

We assume that att=0 the distribution ofu0(x,y,z,w)depends on a single self-similar variable

Then the solution of Eq.(3.5)can be written asX-2U2T=F-1(U),or returning to the original variables,

The formula determines implicitlyuas a function of space coordinates at any moment of timetin the interval 0≤t≤t2.It is worth noticing that this restriction makes it impossible to take the limitt1=t2=0 and to reproduce the solution(3.4).

In what follows,we present some numerical simulations to demonstrate our analytical results(3.7).Here we suppose the initial distribution is as follows：

The pro files of the pulse along thexOyplane and thex-axis are shown in Fig.1 and Fig.2 fort=0.3 and 0.7,respectively.

Figure 1:Pro file of the pulse u(x,y,z,t)for the moment of time t=0.3:(a)along xOy plane;(b)along x-axis.

Figure 2:Pro file of the pulse u(x,y,z,t)for the moment of time t=0.7:(a)along xOy plane;(b)along x-axis.

Remark 3.1.In the present numerical simulation,we have drawn the traveling wave 3-D solutions surfaces and corresponding 2-D solution graphs for the obtained exact solutions of Eq.(3.3).We should stress that all these effects are relevant for finite timet～tpimportant for practical applications.In the study of asymptotical long-time behavior fort?tpwe can put alltpequal to zero,then the wave fronts become asymptotically paraboloidal.

Acknowledgement

The authors would like to express gratitude to the reviewers for pointing out mistakes in the fi rst manuscript and giving many valuable comments.This work was partially supported by NSFC(No.11701244).

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