Research On The Approximate Solutions And Stability Analysis To Several Types Of Nonlinear Equations

Nonlinear partial differential equations can be used to describe various nonlinear wave phenomena in nature.Moreover,they are many applications in other fields,such as fluid mechanic,vibration and control,signal and image processing,computer communication,optics,etc.However,since there is no unified method to obtain all the solutions of nonlinear partial differential equations,it becomes hot topic to discover the new effective methods for solving nonlinear partial differential equations.In this thesis,a variety of methods for finding exact solutions of nonlinear partial differential equations are compared.Based on the Jacobi elliptic functions method,a new and effective method for solving nonlinear partial differential equations—the modified Khater method is proposed.Comparing with eight classic methods(including（G’/G)-expansion method,extended tanhfunction method,Kudryashov method,improved Kudryashov method,improved tan(φ/2)expansion method,novel(G’/G)-expansion method,improved(G’/G)-expansion method),it shows the effectiveness and superiority of the new method.The improved Khater method,the improved extended simple equation,F-expansion method and Jacobi elliptical function method are used to study the nonlinear long-short wave interaction system,the(3+1)-dimensional cubicquintic complex Ginzburg-Landau dynamical equation with broken phase symmetry,the coupled Boiti-Leon-Pempinelli system,the(3+1)-dimensional Kadomtsev-Petviashvili equation,the fractional Kudryashov-Sinelshchikov wave equation and the fractional nonlinear DrinfeldSokolov-Wilson system.Then a bright soliton,dark soliton,breathing soliton,interacting multiple solitons,kink wave soliton,anti-kink wave soliton and trigonometric function solution are obtained.The modulation instability of the related systems is studied by using the standard linear stability analysis method.Furthermore,it derives the dispersion relation and gain function.In this thesis,the solutions,numerical simulation and stability analysis of the complex nonlinear wave system,coupled nonlinear wave system,nonlinear wave system with fractional derivatives are investigated,and many new solutions of related systems have been obtained,which provides a theoretical support for the application of these physical systems.