The Study On Traveling Wave Solutions And Bifurcation Phenomena Of Several Classes Of Nonlinear Equations

In this paper,based on the theory of nonlinear equations,we convert nonlinear equation into linear equation to solve,combined with the theory of linear equations,mainly uses the qualitative theory of dynamical systems and bifurcation method to study（2+1）-dimensional Broer-Kaup-Kupershmit（BKK）equation,（1+1）-dimensional Whitham-Broer-Kaup（WBK）equation,（3+1）-dimensional Generalized Kadomtsev-Petviashvili（GKP）equation,and（2+1）-dimensional Yu-Toda-Sasa-Fukuyama（YTSF）equation.According to the qualitative theory of dynamic systems,find the singular points of the system and determine the type of them,and use the elliptic integral formula to solve the solitary wave solution,blow-up wave solution and periodic wave solution of several equations,Several bifurcation phenomena of kink wave are mainly revealed,and the proof is given by using Maple to simulate the change process.The content of this article is arranged as follows:In the first part,firstly,the basic conclusions of the qualitative theory of dynamical systems are reviewed,and these conclusions are directly quoted in the following texts,and then several research methods of the BKK equation,the WBK equation,the GKP equation and the YTSF equation are introduced.In the second part,the nonlinear wave solution of the BKK equation is solved by using the qualitative theory of dynamical systems and the bifurcation method,and it is revealed that the kink wave can be obtained from the bell-shaped solitary wave,the blow-up wave and the valley-shaped solitary wave branch.In the third part,the traveling wave solution of WBK equation is solved by using qualitative theory of dynamical systems and bifurcation method.In addition to revealing the branching of solitary wave and blow-up wave wave,it is also found that periodic wave can be branched into periodic blow-up wave and line wave,and given detailed proof process.In the fourth part,through two variable substitutions,the GKP equation is transformed into an ordinary differential equation,and an autonomous system is constructed,and the singularity is judged by using the qualitative theory of dynamic systems,and some exact solutions of the GKP equation are obtained through two integrations.In the fifth part,through two variable substitutions,the YTSF equation is transformed into an ordinary differential equation,and an autonomous system is constructed,and the singularity is judged by using the qualitative theory of dynamic systems,and some exact solutions of the YTSF equation are obtained through two integrations.