Elsevier

Chaos, Solitons & Fractals

Volume 26, Issue 4, November 2005, Pages 1189-1194
Chaos, Solitons & Fractals

Homoclinic orbits and periodic solitons for Boussinesq equation with even constraint

https://doi.org/10.1016/j.chaos.2005.02.025Get rights and content

Abstract

In the present paper, we study the explicit homoclinic orbits solutions for “bad” Boussinesq equation with periodic boundary condition and even constraint, and periodic soliton solutions for “good” Boussinesq equation with even constraint.

Introduction

Nonlinear dispersive wave equations provide excellent examples of infinite-dimensional dynamical systems which possess diverse and fascinating solutions describing solitary waves, pattern and singularity formation, dispersive turbulence and the propagation of spatio-temporal chaos. They can be used to illustrate many striking features of nonlinear waves, each of which has been understood by a combination of numerical simulations with the methods of the theory of PDEs and geometric dynamical systems.

In particular, among the solutions of the nonlinear evolutionary equations under spatially periodic boundary conditions there are ones localized (within the period) in space and homoclinic in time (homoclinic orbits). The homoclinic orbits are indicative of chaotic behavior in deterministic nonlinear dynamics. The complete understanding of the homoclinic structure in infinite-dimensional phase is far from being available at present. At the same time, the soliton solutions for nonlinear evolutionary have also widely been studied by lots of researchers, and they are still a hot topic in the theory physics and mathematics [1], [2], [3], [4], [5].

In this paper, we will study Boussinesq equation. The French scientist Joseph Boussinesq (1842–1929) described in the 1870s model equations for the propagation of long waves on the surface of water with a small amplitude. Today one has to distinguish two essentially different Boussinesq (Bq) equations. These equations read as followsutt-uxx-3(u2)xx-uxxxx=0badBqequation,utt-uxx-3(u2)xx+uxxxx=0goodBqequation.

The “good” Bq equation describes the two-dimensional irrotational flow of an inviscid liquid in a uniform rectancular channel. There are known results due to local well-posedness, global existence and blow-up of some solutions [6], [9], [10]. The “bad” Bq equation is used to describe two-dimensional flow of shallow-water waves having small amplitudes. There is a dense connection to the so-called Fermi–Pasta–Ulam (FPU) problem. the existence of Lax pair, Backlund transformation and some soliton-type solutions are known [6], [10].

By the linearized stability analysis, we get the existences of homoclinic orbits for “bad” Bq equation and periodic soliton solutions for “good” Bq equation in this paper. Then by the Hirota’s bilinear method, we get the exact expressions of homoclinic orbits and soliton solutions [7], [8]. These exact expressions of homoclinic orbits are very helpful for the further study of chaotic behavior. At the same time, these homoclinic solutions and periodic soliton solutions are obtained firstly in this research area.

Now we form the main results into the following theorem

Theorem 1

There are homoclinic orbits for “bad” Bq equation and periodic soliton solutions for “good” Bq equation likeu=u0-2[4p2b12+b1p2(eipx+e-ipx)(e-Ωt-γ+b2eΩt+γ)][e-Ωt-γ+b1(eipx+e-ipx)+b2eΩt+γ]2andu=u0-24p2b12+2Ω23p4+Ω2(epx+e-px)cosΩt±Ω23p4+Ω2(epx+e-px)+2cosΩt2,where a, b1, b2, p, Ω satisfy the following relationshipsb2=4p4-(1+6u0)p2Ω2b12,Ω2+(1+6u0)p2-p4=0,Ω=±|p|p2-1-6u0,u0-16.

Section snippets

Linearized stability analysis

Supplementing Boussinesq equation with boundary conditionsu(x,t)=ux+2πp,tt0and even conditionu(-x,t)=u(x,t),where p is a constant.

Now we begin to consider the linearized stability. Firstly, we can see that constant u0 is a solution for Bq equation. Considering a small perturbations of the form for fixed pointu(x,t)=u0[1+ε(x,t)].

Substituting (2.3) into (1.1) and linearizing it, we get the following equationεtt-εxx-6u0εxx±εxxxx=0.

Assuming εt = ν, thenενt=01-k2-6k2u0±k40εν,where εxx = k2ε, then the

Exact homoclinic orbits and soliton solutions

We consider the exact homoclinic orbit solutions for “bad” Bq equation and soliton solutions for “good” Bq equation.

Firstly, involving a transformationu(x,t)=u0+2v(x,t).

The Bq equation can be transformed into the following equationsvtt-(1+6u0)vxx-6(v2)xx-vxxxx=0,vtt-(1+6u0)vxx-6(v2)xx+vxxxx=0.

At first, we consider Eq. (3.2). By the independent variable transformationv=(lnF)xx,(3.2) can be transformed into the following bilinear equationDt2F·F-(1+6u0)Dx2F·F-Dx4F·F-BFF=0where B is an integration

Acknowledgments

This work was supported by NSF of China (No. 10361007) and NSF of Yunnan Provice (No. 2004A0001M).

References (10)

  • N. Ercolani et al.

    Phys D

    (1990)
  • M.J. Ablowitz et al.

    J Comput Phys

    (1996)
  • B.M. Herbst et al.

    Comput Phys Commun

    (1991)
  • V.E. Zakharov

    Collapse of langmuir waves

    Sov Phys JETP

    (1972)
  • M.J. Ablowitz et al.

    SIAM J Appl Math

    (1990)
There are more references available in the full text version of this article.

Cited by (0)

View full text