The Branch And Exact Solutions Of The Schr(?)dinger Equation With Arbitrary Exponents And The Zakharov Equation

Nonlinear partial differential equation is an indispensable branch of modern mathematics,which is an important bridge between mathematical theory and practical application.So far,there is a large number of nonlinear partial differential equations that we need to study in detail.However,there is still no uniform method for solving the exact solutions of partial differential equations.Therefore,it is a very meaningful research for us to find an effective and feasible method.In this thesis,we mainly use the method of dynamical systems,respectively with arbitrary index(2+1)-dimensional nonlinear Schr?dinger equation(NLS)and generalized Zakharov equation of the phase portrait branch are analyzed,and obtain new exact solutions,expanding the solution space.First of all,we need to introduce a traveling wave transform to transform the original partial differential equations into ordinary differential equations,it is equivalent to two-dimensional plane dynamical system.Secondly,combining the qualitative theory of ordinary differential equations,the phase diagram of the system is drawn and the kinetic properties of phase diagrams are discussed.Finally,new exact traveling wave solution of the equation is obtained by integrating the Maple software.At the same time,the solution obtained in this paper is compared with the solution obtained by other methods.It is proved that the obtained solution is new.By studying and solving,we obtain solitary wave solutions,kink wave solutions,periodic wave solutions,anti-kink wave solutions,etc.these solutions are expressed in terms of trigonometric functions,Jacobi elliptic functions and hyperbolic functions.Thus it can be seen that the solution obtained by the dynamic system is rich of the solution space.Compared with other methods,the dynamic system method is more concise and clear,and is worth popularizing.