Travelling Wave Solutions for the Coupled IBq Equations by Using the tanh-coth Method

Based on the availability of symbolic computation, the tanh-coth method is used to obtain a number of travelling wave solutions for several coupled improved Boussinesq equations. The abundant new solutions can be seen as improvement of the previously known data. The obtained results in this work also demonstrate the efficiency of the method. 1. Introduction We consider the following two coupled improved Boussinesq (IMBq) equations of Sobolev type: where and are given nonlinear functions, and are unknown functions, and is a constant that has been derived to describe bidirectional wave propagation in several studies, for instance, in a Toda lattice model with a transversal degree of freedom, in a two-layered lattice model, and in a diatomic lattice. Dé Godefroy has studied (1) as the Cauchy problem under certain conditions and showed that the solution for the Cauchy problem of this system blows up in finite time [1]. Wang and Li have considered the Cauchy problem for (1), proved the existence and uniqueness of the global solution, and given sufficient conditions of blow-up of the solution in finite time by convex methods [2]. The Cauchy problem for (1) has been studied and established the conditions for the global existence and finite-time blow-up of solutions in Sobolev spaces for [3]. For more information, we refer the reader to [3] and references therein. Chen and Zhang have considered the initial boundary value problem for the system of the generalized IMBq type equations: where and are unknown functions, are constants, and and are the given nonlinear functions. As a result, they have proved the existence and uniqueness of the global generalized solution and the global classical solution [4]. Rosenau is concerned with the problem of how to describe the dynamics of a dense lattice via the system where ,？？ are constants, and pointed out that (3) was a convenient vehicle to study the dynamics of a dense lattice [5]. In [6], a transversal degree of freedom was introduced in the Toda lattice. For different order of magnitude of the longitudinal and transversal strains, coupled and uncoupled equations for these fields were derived in the discrete case as well as the continuum limit. The system has been obtained for the longitudinal and transversal strains. and are characteristic lengths in the model, is their ratio (i.e., ), denotes the linear mass density, and is a constant. Besides, travelling wave solutions and numerical solutions of the system have been found. Turitsyn has proved the existence of blow-up for the continuum limit model of the Toda lattice