Solitary Wave and Kink Wave Solutions for a Class of Nonlinear Evolution Equations

By using the bifurcation theory of planar dynamical systems to a class of nonlinear evolution equations, the existence of traveling wave solutions is showed. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.


Introduction
In 1990, Yasunori Nejoh [4] derived the following class of nonlinear evolution equations: under the boundary conditions ,   ,   → 0 as || → 0, where  =  − ,  denotes the velocity of moving frame, and ℘,  1 ,  2 ,  3 ,  4 are arbitrary constant parameters.(1) is a model equation which describes the one-dimensional wave propagation of ion acoustic wave in a maganetised plasma with trapped electrons In this paper, we shall consider the bifurcation behavior of traveling wave solutions of (1) in their parametric spaces and give some solitary wave, kink and anti-kink wave solutions.To find solutions of (1), we set Then, (1) becomes where the parameters  1 ,  2 ,  3 and  4 stand for We consider the following planar Hamiltonian system with the Hamiltonian Clearly, (3) corresponds to the case of the integral constant ℎ = 0. Namely, if we find the dynamics of orbit defined by (, ) = 0, then we obtain the travelling wave solution of (1).
Because we are interesting the existence of the solitary wave and kink wave solutions of (1) by the given boundary conditions.We always assume that the equilibrium (0, 0) is not a center, i.e., the condition  1 ≥ 0 holds.For this aim, the method of bifurcation theory of dynamical systems is very useful (see [1], [2], [3]) in our study.
The rest of this paper is organized as follows.In Section 2, we discuss bifurcations of phase portraits of (5), where explicit parametric conditions will be derived in the case  4 ∕ = 0.In Section 3, we show the existence of solitary wave, kink wave and anti-kink wave solutions of (1) in the case  4 ∕ = 0.

Bifurcation set and phase portraits of the Hamiltonian system (5)
In this section, we shall study the bifurcation set and phase portraits of the Hamiltonian system (5) in the case  4 ∕ = 0. I. Suppose that  4 > 0. In this case, making the parametric transformations (5) becomes which has the Hamiltonian Clearly, on the (, )−phase plane, the abscissas   of equilibrium points (  , 0) of system (7) are the zeros of and zero.Denote that  0 = 0. Notice that which has two zeros at Let (  , 0) be an equilibrium point of (7).At this point, the determinant of the linearized system of (7) has the form Suppose that where   ,  = 1, 2, 3 are real numbers.1.  = 0 i.e.,  2 = 0.In this case,  * 1 = 0,  * 2 = − 8 15 .On the (, )−upper-half parametric plane, there are three bifurcation curves: By the theory of planar dynamical systems, we know that if (  , 0) > 0 or (< 0), then the equilibrium (  , 0) is a center (or a saddle point); if (  , 0) = 0 and the Poincare index of (  , 0) is zero, then (  , 0) is a cusp.By using the above facts to do qualitative analysis, under different parameter conditions, we obtain the following bifurcations of phase portraits of system (7) shown in Fig. 1. x (1-5) (1-6) Fig. 1 The bifurcation set and bifurcations of phase portraits of (7) for  2 = 0 .

Solitary wave and kink wave solutions of (1).
For the case  4 ∕ = 0, it is not possible to obtain exact explicit parametric representation of solitary wave and kink wave solutions of (1).From the discussion of section 2, we have the following results.