Topological Soliton Solution and Bifurcation Analysis of the Klein-Gordon-Zakharov Equation in(1+1)-Dimensions with Power Law Nonlinearity

Abstract

This paper addresses the Klein-Gordon-Zakharov equation with power law nonlinearity in (1+1)-dimensions. The integrability aspect as well as the bifurcation analysis is studied in this paper. The numerical simulations are also given where the finite difference approach was utilized. There are a few constraint conditions that naturally evolve during the course of derivation of the soliton solutions. These constraint conditions must remain valid in order for the soliton solution to exist. For the bifurcation analysis, the phase portraits are also given.

Exact traveling wave solutions to the (3+1)-dimensional mKdV–ZK and the (2+1)-dimensional Burgers equations via exp(−Φ(η))-expansion method

“…Many powerful methods to seek for exact solutions to the nonlinear partial differential equations have been proposed. Among these are the direct algebraic method [1], the Lie symmetry group method [2], the inverse scattering transform [3], the complex hyperbolic function method [4,5], the rank analysis method [6], the ansatz method [7][8][9][10][11][12][13][14][15][16][17][18], the ðG 0 =GÞ-expansion method [19][20][21][22][23][24][25][26], the modified simple equation method [27], the exp-functions method [28], the sine-cosine method [29], the Jacobi elliptic function expansion method [30,31], the F-expansion method [32], the Backlund transformation method [33], the Darboux transformation method [34], the homogeneous balance method [35][36][37], the Adomian decomposition method [38,39], the auxiliary parameter method [40], the homotopy perturbation method [41][42][43], the expðÀuðgÞÞ-expansion method [44]…”

“…A large number of literature [9][10][11][12][13][14][15][16][17][18], where the various types of nonlinear mKdV-ZK equation and Burgers are studied and have demonstrated their analytical solutions as well as traveling wave solutions using different methods. For instance, Khan et al [47] have described the new soliton solutions of the generalized Zakharov equations using He's variational approach.…”

The Klein–Gordon–Zakharov equations with the positive fractional power terms and their exact solutions

“…Due to their potential application in plasma physics, the nonlinear Klein-Gordon-Zakharov system has been paid attention by many researchers. some exact solutions for the Zakharov equations are obtained using different methods [4][5][6][7][8][9][14][15]. In the Ref.…”

New exact traveling wave solutions to the (1+1)-dimensional Klein-Gordon-Zakharov equation for wave propagation in plasma using the exp(-Φ(ξ))-expansion method

“…The solutions of this equation play a vital rule to analyze the wave propagation of various types of physical phenomena in the related fields. There is an amount of paper [35][36][37][38][39][40][41][42][43][44][45][46], where the various types of nonlinear KGZ equation have been studied. Some of the KGZ equations are also appeared to describe the acoustic wave propagation in plasma physics.…”