Bifurcations of travelling wave solutions in variant Boussinesq equations

Abstract

The bifurcations of solitary waves and kink waves for variant Boussinesq equations are studied by using the bifurcation theory of planar dynamical systems. The bifurcation sets and the numbers of solitary waves and kink waves for the variant Boussinesq equations are presented. Several types explicit formulas of solitary waves solutions and kink waves solutions are obtained. In the end, several formulas of periodic wave solutions are presented.

Analytical Solutions for the Elastic Circular Rod Nonlinear Wave, Boussinesq, and Dispersive Long Wave Equations

“…Substituting the second formula of (7) into the first one and assuming 1 = 1 − 2 , 2 = (3/2) , 3 = −(1/2), = 1 + 1 , we can obtain (3). So the solutions of (7) are equivalent to that of (3) and the second formula of (7).…”

“…The variant Boussinesq equations were discussed in [7]. They are coupling wave equations which can be expressed as follows:…”

“…Obviously, the first formula of (10) and (3) have the same characteristics. On the other hand, the solutions of (10) and (7) are same with each other except for the coefficients. Therefore, in following parts, the solutions of (5) are presented.…”

“…In recent years there has been much interest in some variants of the Boussinesq systems [2][3][4][5][6]. These coupled Boussinesq equations [7] arise in shallow water waves for two-layered fluid flow. This situation occurs when there is an accidental oil spill from a ship which results in a layer of oil floating above the layer of water.…”

Solutions for the Finite Deformation Elastic Rod Nonlinear Wave Equation

“…In recent years there has been much interest in some variants of the Boussinesq systems. These coupled Boussinesq equations [2] arise in shallow water waves for two layered fluid flow. Over the last year, many significant methods have been developed to find more exact solutions.…”