Bifurcation analysis and the travelling wave solutions of the Klein–Gordon–Zakharov equations

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.…”

“…Using the solitary wave Ansatz method, 1-soliton solution of the Klein-Gordon-Zakharov equation with power law nonlinearity is given, and the numerical simulations are included that supports the analysis [7]. Bifurcation analysis and the travelling wave solutions of the Klein-Gordon-Zakharov equations are studied [8], and topological soliton solution of the Klein-Gordon-Zakharov equation in (1+1)-dimensions with power law nonlinearity are derivrd and bifurcation analysis are studied in Ref. [9].…”

Exact Solutions for a Class of Nonlinear PDE with Variable Coefficients Using ET and ETEM

“…Especially in the multifarious real physical background such as the field of nonlinear optical crystal and plasma, nonlinear partial differential equations (PDEs) with variable coefficients can often provide realistic and powerful models than the corresponding constant coefficients ones when the inhomogeneities of media are considered. There are lots of studies have been conducted with different types of the PDEs, such as the modified trigonometric functions series [1] [2], the G G ′ expand method [3] [4], the first integral method [5] [6], the modified CK direct method [7] [8] [9], the ( ) ( ) method [14] [15], the Lie symmetry analysis method [16]- [21] and the extended trial equation method [22] [23]. In this paper, we will use the modified CK direct method and the extended trial equation method (ETEM) to discuss the exact solutions for the following generalized fifth-order nonlinear partial differential KDV equation [24] Sawada-Kotera (SK) equation: when Lou modified the CK direct method and proposed a simple method which called the modified CK direct method.…”

Bifurcation Analysis and Solutions of a Higher-Order Nonlinear Schrödinger Equation

“…In this paper, with the aid of Mathematica, we study the new traveling wave solutions for a higher-order NLS equation that contains the non-Kerr nonlinear terms, which describes propagation of very short pulses in highly nonlinear optical fibers by using different elliptic functions. The bifurcation theory method is widely used to solve differential equations [9][10][11][12]. By using this method of dynamical systems, we obtain the explicit expressions of the bounded traveling wave solutions for the equation and investigate the relation between the bounded orbit of the traveling wave system and the energy level ℎ.…”