The Exact Solutions Of GDNLSE And Generalized DS Equation With Arbitrary Power Nonlinearities

The nonlinear partial differential equation(PDE)is an indispensable branch of modern mathematics,which is involved in almost every field of natural science and social science.It is often attributed to the problem of solving nonlinear partial differential equations.However as yet,there is no unified and universal method to solve the nonlinear partial differential equations.Therefore,it is still a very important and valuable work to find an effective and feasible method for finding the exact solutions to partial differential equations.In this thesis,we use the dynamic system method and the first integral method to find the exact traveling wave solutions of the generalized derivative nonlinear Schrodinger equation(GDNLSE)and the generalized Davey Stewart-son(DS)equation with arbitrary nonlinear terms.By using the dynamic system method,firstly the proper traveling wave transformation is introduced to trans-form the partial differential equation into ordinary differential equation,which is equivalent to a planar dynamic system.Secondly,we use Maple software and com-bine the qualitative theory of ordinary differential equation to obtain the phase portrait of the system and to discuss the dynamic properties of the phase portrait.Finally,we obtain the exact traveling wave solutions of the equation based on the phase orbit in the phase portrait by Maple software.Using the first integration method,the parameter space is discussed according to the division theorem of polynomial function,and the exact solutions of the equation are obtained.At the same time,the solutions obtained in this thesis are compared with those obtained by other methods.It shows that the solutions obtained in this thesis are both generalized and extended.Through the study,we get the exact solutions including the solitary wave solution,the kink wave solution,the periodic wave solution and the anti-kink wave solution.They are represented by trigonometric functions,exponential functions,Jacobi elliptic functions and hyperbolic functions.The results obtained by the dynamic system method and the first integral method are more diverse.From the solving process,the dynamic system method has the characteristics of simplicity,directness and effectiveness,and it also shows the effectiveness of solving nonlinear partial differential equations by using the first integral method.