Separation Transformation and a Class of Exact Solutions to the Higher-Dimensional Klein-Gordon-Zakharov Equation

Abstract

The separation transformation method is extended to then+1-dimensional Klein-Gordon-Zakharov equation describing the interaction of the Langmuir wave and the ion acoustic wave in plasma. We first reduce then+1-dimensional Klein-Gordon-Zakharov equation to a set of partial differential equations and two nonlinear ordinary differential equations of the separation variables. Then the general solutions of the set of partial differential equations are given and the two nonlinear ordinary differential equations are solved by extendedF-expansion method. Finally, some new exact solutions of then+1-dimensional Klein-Gordon-Zakharov equation are proposed explicitly by combining the separation transformation with the exact solutions of the separation variables. It is shown that, for the case ofn≥2, there is an arbitrary function in every exact solution, which may reveal more nontrivial nonlinear structures in the high-dimensional Klein-Gordon-Zakharov equation.

Orbital stability of solitary waves of the coupled Klein–Gordon–Zakharov equations

“…(13), we obtain for c ¤˙1 From (11), we obtain q D !c 1 c 2 . Thus, combining (14), Eq. (10) becomes…”

“…Furthermore, from (8), (14), and (17), we can obtain the solitary wave solution u.x, t/ and v.x, t/ of Eq. (1).…”

“…Chen and Zhang proposed two new different schemes for an initial‐boundary‐value problem of the coupled KGZ equations and proved stability and convergence of different solutions. Many authors obtained some explicit exact solitary wave solutions and numerical solutions for the coupled KGZ equations by using a various of different approaches ( and references therein). Note that if α = 1 and β = 0, then system reduces to the classical KGZ equations The system arises in the study of the interaction between a Langmuir wave and an ion sound wave in plasma.…”