Non-integrable variants of Boussinesq equation with two solitons

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Abstract

Three variants of the Boussinesq equation, namely, the (2 + 1)-dimensional Boussinesq equation, the (3 + 1)-dimensional Boussinesq equation, and the sixth-order Boussinesq equation are studied. The Hirota bilinear method is used to construct two soliton solutions for each equation. The study highlights the fact that these equations are non-integrable and do not admit N-soliton solutions although these equations can be put in bilinear forms.

Introduction

It is well known that completely integrable equations give rise to N-soliton solutions for finite N, where N  1. All completely integrable evolution equation can be put into bilinear forms. However, the converse is not true; many non-integrable equations can be put into bilinear forms. Multiple-soliton solutions of completely integrable evolution equations can be obtained by different methods, such as, the inverse scattering method, the Bäcklund transformation method, the Hirota bilinear method, and many others [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. Each of these methods has its features, but the Hirota’s bilinear method is rather heuristic and gives multiple soliton solutions for a wide class of nonlinear evolution equations in a systematic way.

The (1 + 1)-dimensional Boussinesq equationutt-uxx+α(u2)xx-uxxxx=0,α0,is completely integrable and admits multiple soliton solutions. The Boussinesq equation describe gravity-induced surface waves as they propagate at a constant speed in a canal of uniform depth.

Many variants of the Boussinesq equation were derived to describe physical phenomena. In this work we will study three variants of the Boussinesq equation, namely, the (2 + 1)-dimensional Boussinesq equation [1], [2], [3], [4], [5]utt-uxx-uyy++α(u2)xx-uxxxx=0,α0,the (3 + 1)-dimensional Boussinesq equation [3], [5]utt-auxx-buyy-cuzz+α(u2)xx-uxxxx=0,α0,and the sixth-order Boussinesq equationutt-uxx-15uu4x+30uxu3x+15(u2x)2+45u2u2x+90uux2+u6x=0.

Eq. (2) is introduced to describe the propagation of gravity waves on the surface of water, in particular the head-on collision of oblique waves [1]. It combines the two-way propagation of the classical Boussinesq equation with the weak dependence on a second spatial variable y. Eq. (3) combines the two-way propagation of the classical Boussinesq equation with the weak dependence on two other spatial variables y and z. The sixth-order Boussinesq equation (4) was formally derived in [3] by using the extension of the bilinear forms of the classical Boussinesq equation.

The majority of the works in solitary waves theory focused on the integrability of some of the nonlinear evolution equations and to derive the N-soliton solutions of these equations. The investigation of non-integrable equations is rare compared to the aforementioned studies. The aim of this work is to apply a combination of Hirota’s method [6], [7], [8], [9], [10], [11], [12], [13], [14], [15] and Hereman’s method [16], [17], [18] to show that Eqs. (2), (3), (4) are non-integrable and each gives rise to two soliton solutions only. It was thought in [3] that Eq. (4) is integrable, but it was corrected in the erratum [4] that it is a non-integrable equation, and this will be confirmed here.

Section snippets

The (2 + 1)-dimensional Boussinesq equation

We begin our study on the (2 + 1)-dimensional Boussinesq equationutt-uxx-uyy+α(u2)xx-uxxxx=0,α0.This equation would seem to be the archetype for waves that propagate in opposite direction in (2 + 1)-dimensions [1].

Using the bilinear differential operator, Eq. (5) has the bilinear formDt2-Dx2-Dy2-Dx4(f·f)=0,where f(x, y, t) is the auxiliary function. The bilinear form (6) can be written in terms of f byf(ftt-fxx-fyy-fxxxx)-ft2-fx2-fy2-4fxfxxx+3fxx2=0.To achieve our goal of this work, we will employ a

The (3 + 1)-dimensional Boussinesq equation

We next study the (3 + 1)-dimensional Boussinesq equation [5]utt-auxx-buyy-cuzz+α(u2)xx-uxxxx=0,α0.Eq. (20) combines the two-way propagation of the classical Boussinesq equation with the weak dependence on two other spatial variables y and z.

Using the bilinear differential operator, Eq. (20) has the bilinear formDt2-aDx2-bDy2-cDz2-Dx4(f·f)=0,where f(x, y, z, t) is the auxiliary function.

To achieve our goal for this equation, we will proceed as before. Substitutingu(x,y,z,t)=eθi,θi=kix+riy+siz-ωit,

The sixth-order Boussinesq equation

We close our discussion by considering the sixth-order Boussinesq equationutt-uxx-15uu4x+30uxu3x+15(u2x)2+45u2u2x+90uux2+u6x=0.The bilinear form for this equation is given by [3]Dt2-Dx2-Dx6f·f=0.Proceeding as in the preceding two sections, we substituteu(x,t)=eθi,θi=kix-ωit,into the linear terms of the sixth-order Boussinesq equation (36) gives the dispersion relationωi=-ki1+ki4,i=1,2,,N,and hence θi becomesθi=kix+ki1+ki4t.This means thatf1(x,t)=eθ1=ek1x+k11+k14t.To determine R, we substituteu(

Discussion

A combination of Hirota’s method and Hereman’s method were used to formally study three variants of the Boussinesq equations. The three variants fail to belong to the class of completely integrable equations. The obtained conjecture was based on the fact that only two soliton solutions were derived for each variant. It is important to point out that, although all three variants are represented by bilinear forms, but these variants were proved to be non-integrable. The Hirota’s bilinear method

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