Construction Of The Exact Solutions For Nonlinear Schr(?)dinger Equation And Ginzburg–Landau Equation

The unstable nonlinear schr(?)dinger equation and the space–time fractional complex Ginzburg–Landau equation have been studied by the analysis method for dynamical systems,symbolic operation method and complete discrimination system for polynomial method.The main objects,methods and results are present as follows:1? By using the envelope wave transformation,the unstable nonlinear schr(?)dinger equation have been transformed into a plane dynamic system,and the dispersion relation,Hamiltonian of the system have been obtained;with the aid of analysis method for dynamical systems,the equilibrium point and classification of the system and the branch phase diagram of the system under different parameters have been obtained.By Integrating along different evolutionary orbits;the exact traveling wave solutions of the nonlinear unstable Schrodinger equation have been constructed;which including periodic traveling wave solution,torsional wave solution,periodic blasting wave solution,and bell solitary wave solution.2?The space–time fractional complex Ginzburg–Landau equation have been converted into an ordinary differential equation by use of integrated fractional–order derivative and the fractional complex transformation,then the equation is reduced to an elementary integral form.Then,the roots of the polynomials in the integrand function are classified by the complete discriminant system of polynomials,and so a series of exact solutions have been obtained,which includes solitary wave solutions,rational solution,triangular function periodic solution,elliptic functions double periodic solutions.3?Transform the unstable nonlinear Schr(?)dinger equation into ordinary differential equations by traveling wave transformation,separate the real and imaginary parts and set them to zero,and then use the symbolic operation method to construct a series of exact solutions,the solutions including bell-shaped soliton solutions,trigonometric function solutions,rational function solution,twisted wave solution and Jacobi elliptic function solution.