Solving Three Kinds Of Nonlinear Differential Equations:New Method For Rogue Wave Solutions And New Solutions

Solving nonlinear differential equations is not only a research topic in nonlinear science,but also an important research direction in soliton theory.Although there are some methods in the field of research,it is very meaningful to find effective methods for solving nonlinear differential equations and obtain new solutions of nonlinear differential equations.Rouge wave solutions are a special kind of solutions of nonlinear evolution equations,whose propagation is characterized by suddenness and large amplitude.The methods for constructing rouge wave solutions include Darboux transformation,inverse scattering transformation,bilinear method and algebraic geometry reduction method,etc.There are two starting points in this paper: one is to give a new method to construct rouge wave solutions of nonlinear partial differential equations with complex coefficients by using the ansatz form of complex multiple rational exponential functions;the other is to find some new solutions of nonlinear differential equations based on auxiliary ordinary differential equation method.The main work of this dissertation is as follows:Firstly,the complex multiple rational exponential functions ansatz is divided into three cases: solitary wave ansatz solution,N-wave ansatz solution and rouge wave ansatz solution for constructing different types of exact solutions of nonlinear partial differential equations with complex coefficients.The solitary wave ansatz solution is used to solve the real part and imaginary part of the complex coefficient equation after separating variables,thereby construct the solitary wave solution of the complex-coefficient equations indirectly;the N-wave ansatz solution is directly used to construct single-wave solution,double-wave solution and three-wave solution of the complex-coefficient equations,and a formula of N-wave solution is concluded;the rouge wave ansatz solution is directly applied to the complex-coefficient equations in order to find a new method to construct the rouge wave solutions.To test the validity of the three cases of the complex multiple rational exponential function ansatz,a nonlinear Schr(?)dinger equation with variable coefficients is selected as a positive example,and the spatial structures and dynamical evolutions of the three kinds of solutions are simulated.Secondly,the auxiliary ordinary differential equation method is extended to the nonlinearSchr(?)dinger equation with variable coefficients,Whitham-Broer-Kaup(WBK)equation with variable coefficients and fractional nonlinear oscillation equation,and many new exact solutions are obtained,including hyperbolic function solutions,trigonometric function solutions and rational solutions.The simulations of the solutions show that the nonlinear oscillations generated by the amplitudes in the dynamic evolution process are not only influenced by the coefficient functions,but also by the oscillation functions,noises and fractional orders.