Research On Nonlinear Schr(?)dinger Equations And Fractional Klein-Gordon Equation

Nonlinear partial differential equation is an important mathematical model,which is widely used in many fields such as epidemiology,fluid mechanics,signal processing,statistical analysis,and control theory.Therefore,it is particularly important to analyze the evolution law of equations and construct the exact solutions of equations.In this thesis,nonlinear Schr (?)dinger equations and fractional Klein-Gordon equation are analyzed by two methods.The specific research objects,methods and main results are as follows:1.The generalized nonlinear Schr(?)dinger equation is transformed into an elementary integral by means of traveling wave transformation,and then a series of explicit solutions are constructed by using the complete discriminant method of polynomials.2.The generalized quintic derivative nonlinear Schr (?)dinger equation is studied by using the complete discriminant system method of polynomials.A series of implicit and explicit solutions are constructed including thirteen new exact solutions.3.The space-time fractional Klein-Gordon equation is transformed into a plane dynamic system by the bifurcation method of dynamic system,and the plane phase diagram is drawn.From the orbit in the graph,five kinds of explicit solutions are gained.