Persistence Of The Solitary Wave Of The Perturbed Cubic-Quintic Nonlinear Schrodinger Equation With Higher-Order Dispersion Term

In this paper,the persistence of solitary waves of two perturbed cubic-quintic nonlinear Schrodinger equations with higher-order dispersion term is researched.One is the cubic-quintic nonlinear Schrodinger equation with fourth-order dispersion term and another is the cubic-quintic nonlinear Schrodinger equation with third-order dispersion term They are nonlinear mathematical models describing the propagation characteristics of optical solitons in an ultrashort optical pulse fiber.Firstly,the solitary wave of the cubic-quintic nonlinear Schrodinger equation with fourth-order dispersion term after periodic perturbation and damping perturbation is studied the perturbed nonlinear Schrodinger equation are simplified into ordinary differential equations by using the traveling wave transformation and substitution method.When ε=0,the equilibrium points of systems and the corresponding phase diagrams are analyzed and classified,The solitary wave solutions are obtained.When ε≠0,the chaos threshold in the sense of Smale horseshoes is obtained by Melnikov’s method.Secondly,the solitary wave of the cubic-quintic nonlinear Schrodinger equation with third order dispersion term after linear loss perturbation,damping perturbation,periodic perturbation and second order dispersion perturbation is studiedBy using traveling wave transformation and multi-scale transformation,the unperturbed nonlinear Schrodinger equation is transformed into a 5-order ordinary differential equation system with decoupling.The 5-order ordinary differential equations are reduced to 4-order ordinary differential equations by the method of section limitation.When ε=0,invariant manifolds of unperturbed systems are analyzed and homologous manifolds are given.Whenε≠0,the monopulse Melnikov function is obtained by high dimensional Melnikov method,and the chaos threshold in the sense that the system may produce Smale horseshoe is also obtained.In addition,we select the perturbation parameters as variables and perform numerical simulations to obtain the chaotic threshold map of the system.The maximum Lyapunov exponential map and the bifurcation map are used to verify the chaotic behavior of the system within the chaotic threshold obtained by Melnikov’s method.So the parameter region,where the solitary waves can remain stable in long distance propagation,can be obtained by avoiding this chaotic threshold.