Study On Dynamical Systems And Exact Solutions Of Generalized Fourth-order Dispersion Nonlinear Schrodinger Equation

As an important branch of nonlinear science,bifurcation theory is a powerful method to analyze the structural stability mechanism of nonlinear dynamic system.It has penetrated into all branches of mathematics and plays a very important role.In this paper,the exact solutions and dynamic properties of generalized fourth order dispersion nonlinear Schrodinger equation are studied by using the bifurcation theory of dynamic system.Firstly,based on the bifurcation theory of plane dynamic system,six phase portraits of ordinary differential system corresponding to generalized fourth order dispersion nonlinear Schrodinger equation are constructed.The existence of the bright solitary wave,dark solitary wave,periodic wave,periodic breaking wave and unbounded wave are qualitatively revealed.Secondly,the quantitative relationship between phase orbits and the energy level h is established,and the conditions for the generation of different types of solutions are given.Finally,the exact solutions of phase orbits are given by using the elliptic equation and the first integral.In addition,the generation mechanism of the rogue wave solution in the fourth order dispersion nonlinear Schrodinger equation is studied under integrable conditions.According to the first-order exact breather solution of the integrable equation over a nonvanishing background,the expressions of group velocity and phase velocity are derived.The jump phenomenon of group velocity and phase velocity is found at a certain point by limit analysis.Furthermore,by limiting the parameters of the breather solution at the jumping point,the first-order rogue wave solution is obtained,which proves that the rogue wave can be transformed from the breather solution at the jumping point.It also shows that the discontinuity of the structure of the equation will lead to the rogue wave.