Traveling Wave Solutions Of Several Types Of Nonlinear Partial Differential Equations

The nonlinear wave equation is proposed to describe various nonlinear wave phenomena in most of branches of physics such as fluid dynamics,plasma physics,solid state physics,condensed matter physics.It is of great significance to study its traveling wave solutions,which not only helps to understand various complex nonlinear wave phenomena and wave propagation,but also can be used to verify the correctness of numerical solutions.In this paper,we first study the exact traveling wave solutions for three kinds of nonlinear wave equations,including(2+1)-dimensional generalized dissipative Ablowitz-Kaup-Newell-Segur(AKNS)equation,Bogoyavlenskii-Kadomtsev-Petviash-vili(BKP)equation and(3+1)-dimensional Jimbo-Miwa equation.The corresponding traveling wave systems of these three nonlinear equations can be transformed into the dynamic systems inR~3 which also contain a two-dimensional invariant manifold.Fortunately,these invariant manifolds not only determine most of the dynamical behavior of the original system,but is also conservative,which allows us to start with them.With the bifurcation method of dynamic system,we investigate the phase space geometric features of their two-dimensional invariant manifolds under different parameter bifurcation sets systematically and detailedly,and obtain three kinds of bounded traveling wave solutions of the(2+1)-dimensional generalized dissipative AKNS equation by calculating complex elliptic integrals.In particular,we also give the exact expressions and corresponding existence conditions of all bounded and unbounded traveling wave solutions for the BKP equation and the(3+1)-dimensional Jimbo-Miwa equation without any omission.Furthermore,we also study the ZK(n,-n,2n)equation.Different from the first three types of nonlinear equations,the traveling wave system of ZK(n,-n,2n)equation is only a dynamical system inR~2,but its defined vector field has a straight line,which makes it no longer analytic.This will lead to strange phenomena near the straight line,and it will be very difficult to study orbits around the straight line,which needs us to use more complicated techniques to analyze it meticulously.In this paper,we overcome these difficulties and use the theory of the singular traveling wave system and the bifurcation theory of dynamical system to analyze in detail the changes and regularities of the topological structure of the orbital family of the corresponding phase spaces of the ZK(n,-n,2n)equation with the change of parameters,and then obtain 13 very complex parameter bifurcation sets.With the help of Maple,we show 63 kinds of phase portraits under different parameter bifurcation sets.Using the geometric spaces of those obtained phase portraits and calculating the complex elliptic integrals along each bounded orbit,we give the exact expressions of the 37 bounded traveling wave solutions of the ZK(n,-n,2n)equation one by one.