Analytical Study Of Some Nonlinear Evolution Equations

Nonlinear science has always been one of the hot spots in the research of scholars,this is determined by the wide applicability of nonlinear science.Both in scientific research and in practical application,nonlinear model has played a very important role.However,because of the diversity and complexity of the nonlinear science,there are no fixed methods for solving the nonlinear evolu-tion equation.The analytical solutions of some nonlinear evolution equations are studied in this paper.The main work of this paper is as follows four aspects:(1)In the first chapter,introducing the main research background,research methods and techniques as well as some common methods for solving nonlinear evolution equations mainly.(2)In the second chapter,the reaction-diffusion equations are studied and discussed in detail by using bifurcation theory,firstly,the two reaction diffusion equation is studied,the equation is changed into plane system,then discuss the phase trajectory and the equilibrium points of the system and the corresponding types.The equation is solved by using the method of variable separation,get the analytical solution after the solution and the image are discussed and gives the corresponding relationship.Discussed the characteristics of the extened reaction diffusion equation and it's plane system,analysis the equilibrium point and it's phase portrait.(3)In the third chapter,the variable coefficient 3+1 dimensional coupled NLS equation is studied.Firstly,introducing the physical background of the variable coefficient 3+1 dimensional coupled NLS equation,then obtain the new bilinear form of the equation by Hirota method and symbolic computation.Secondly,we obtain the single soliton solution and two soliton solution of the equation with small parameter.Then the Backlund transform of the equation is achieved.Finally,the interaction between the soliton solutions is achieved by the image analysis.(4)The last chapter is the summery and prospects.