Homoclinic orbits and periodic solitons for 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 follows
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 likeandwhere a, b1, b2, p, Ω satisfy the following relationships
Section snippets
Linearized stability analysis
Supplementing Boussinesq equation with boundary conditionsand even conditionwhere 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 point
Substituting (2.3) into (1.1) and linearizing it, we get the following equation
Assuming εt = ν, thenwhere 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 transformation
The Bq equation can be transformed into the following equations
At first, we consider Eq. (3.2). By the independent variable transformation(3.2) can be transformed into the following bilinear equationwhere B is an integration
Acknowledgments
This work was supported by NSF of China (No. 10361007) and NSF of Yunnan Provice (No. 2004A0001M).
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