Exact Traveling Wave Solutions Of (3+1)-Dimensional Plasma Equations

Nonlinear partial differential equations are widely used in many fields,while the study of the traveling wave solutions for nonlinear partial differential equations can help people to understand some natural phenomenon better.Therefore,it is of great significance to study the exact traveling wave solutions of nonlinear partial differential equations.In this paper,the exact traveling wave solutions of(3+1)-dimensional plasma equations are constructed by the generalized Riccati equation mapping method,the exp(-Φ(ξ))expansion method and the Riccati equation expansion method,the specific plasma equations have the following forms:1.The(3+1)-dimensional modified KdV-Zakharov-Kuznetsov equation ut+αu2ux+uxxx+uxyy+uxzz=0;(Ⅰ)2.The(3+1)-dimensional Kadomtsev-Petviashvili equation(ut+6uux+uxxx)x-3uyy-3uzz=0;(Ⅱ)3.The(3+1)-dimensional Jimbo-Miwa equation uxxxy+3uyuxx+3uxuxy+2uyt-3uxz-0.(Ⅲ)By studying the traveling wave solutions of the above equations(Ⅰ,Ⅱ,Ⅲ),the following conclusions are obtained:First,by using the traveling wave transformation to the equation(Ⅰ),an equivalent third order ordinary differential equation is obtained,a corresponding second order ordinary differential equation is obtained after integral operation once.With the help of theory and method of planar dynamical system,the equivalent planar dynamical system is given and analyzed qualitatively,there exsit two bell solitary wave solutions,two kink solitary wave solutions and countless periodic solutions for the equation(Ⅰ).The exact expressions of kink solitary wave solutions,singular kink solitary wave solutions and periodic solutions of the equation(Ⅰ)are obtained by using the generalized Riccati equation mapping method.The behavior of kink solutions and periodic solutions of the equation(Ⅰ)are given intuitively by Matlab.Second,by using the traveling wave transformation to the equation(Ⅱ),an equivalent fourth order ordinary differential equation is obtained,a corresponding second order ordinary differential equation is obtained after integral operations twice.With the help of theory and method of planar dynamical system,the equivalent planar dynamical system to this second order ordinary differential equation is analyzed qualitatively,there exsit two bell solitary wave solutions and countless periodic solutions for the equation(Ⅱ).The exact expressions of bell solitary wave solutions,periodic solutions and nonsmooth travaling wave solutions of the equation(Ⅱ)are obtained by using the exp(-Φ(ξ))expansion method.The behavior of bell solitary wave solutions,periodic solutions and nonsmooth travaling wave solutions of the equation(Ⅱ)are given intuitively by Matlab.Finally,by using the traveling wave transformation and integral operation to the equation(Ⅲ),a corresponding third order ordinary differential equation is obtained.The exact expressions of kink solitary wave solutions,singular kink solitary wave solutions,periodic solutions and singular solutions of the equation(Ⅲ)are obtained by using the Riccati equation expansion method.The behavior of kink solutions,singular solutions and periodic solutions of the equation(Ⅲ)are given intuitively by Matlab.