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This paper addresses the Klein-Gordon-Zakharov equation with power law nonlinearity in (

The theory of nonlinear evolution equations (NLEEs) has come a long way in the past few decades [

The integrability aspects and the bifurcation analysis will be the main focus of this paper. The ansatz method will be applied to obtain the topological 1-soliton solution, also known as the shock wave solution, to this equation. The constraint conditions will be naturally formulated in order for the soliton solution to exist. Subsequently, the bifurcation analysis will be carried out for this paper. In this context, the phase portraits will be given. Additionally, other traveling wave solutions will be enumerated. Finally, the numerical simulation to the equation will be given. The finite difference scheme will also be given.

The KGZ equation with power law nonlinearity in

Finally, equating the coefficients of the linearly independent functions

This section will carry out the bifurcation analysis of the Klein-Gordon-Zakharov equation with power law nonlinearity. Initially, the phase portraits will be obtained and the corresponding qualitative analysis will be discussed. Several interesting properties of the solution structure will be obtained based on the parameter regimes. Subsequently, the traveling wave solutions will be discussed from the bifurcation analysis.

We assume that the traveling wave solutions of (

Substituting (

Substituting (

If

If

If

Therefore, we obtain the bifurcation phase portraits of system (

The bifurcation phase portraits of system (

Let

Next, we consider the relations between the orbits of (

Set

According to Figure

Suppose that

Suppose that

(I) When

(II) When

From the qualitative theory of dynamical systems, we know that a smooth solitary wave solution of a partial differential system corresponds to a smooth homoclinic orbit of a traveling wave equation. A smooth kink wave solution or a unbounded wave solution corresponds to a smooth heteroclinic orbit of a traveling wave equation. Similarly, a periodic orbit of a traveling wave equation corresponds to a periodic traveling wave solution of a partial differential system. According to the above analysis, we have the following propositions.

If

If

Firstly, we will obtain the explicit expressions of traveling wave solutions for (

Completing the above integrals we obtain

Secondly, we will obtain the explicit expressions of traveling wave solutions for (

Completing the above integrals we obtain

We decompose the function

To get the numerical solution the initial conditions are taken from the exact solution (

Topological solution for the Klein-Gordon-Zakharov equations.

This paper studied the KGZ equation in

These results are pretty complete in analysis. They are going to be extended in the future. An obvious way to expand or generalize these results is going to extend to