Research On Darboux Transformation,Inverse Scattering Transformation And Analytical Solutions Of Several Nonlinear Evolution Equations

Due to the importance of nonlinear evolution equations,the research of its analytical solution occupies an important position in the field of nonlinear science,so the main purpose of this thesis is to develop the existing Hirota bilinear method,Darboux transformation method and Riemann-Hilbert approach to study analytical solutions of several kinds of nonlinear partial differential equations.The relationship between the parameters of the obtained analytical solutions and the phenomena are studied through image simulation,and we find some interesting dynamical behaviors of nonlinear evolution equations;in addition,we discovered some interesting transition mechanisms of nonlinear waves by developing characteristic lines and phase shift analysis.The main contents of this thesis are as follows:In Chapter 1,the research background and common research methods of integrable systems are briefly introduced,and the basic theoretical knowledge of solitons is briefly introduced.In addition,the main research contents of this work are given.In Chapter 2,by developing the existing bilinear method and the long wave limit method,some interesting analytical solutions to the(2+1)-dimensional generalized Bogoyavlensky-Konopelchenko(gBK)equation,including Nth-order breather solutions and Nth-order lump solutions,have been discovered for the first time.In addition,we give the semi-rational solutions of the equation and reveal the interaction between the lump solution,the soliton solution and the breather solution.In Chapter 3,based on the characteristic line analysis,the dynamic properties of the transformed nonlinear waves of the variable coefficient Caudrey-Dodd-GibbonKotera-Sawada equation are studied for the first time.Firstly,the existing bilinear method is developed to give some interesting analytical solutions of the equation,including N-soliton solution,breather wave solution,lump solution,mixed solution and second-order breather wave solution.Through the characteristic line analysis,the second-order breather wave solution of the collision mode and the collision-free mode are discussed.After that,we give the conversion conditions of the breather wave,and analyze various nonlinear wave structures.Furthermore,we study the gradient properties,superposition mechanism,formation mechanism,local properties and oscillation characteristics of the transformed nonlinear wave.Based on the geometric property of the characteristic line,we analyze the time-varying characteristics of the transformed nonlinear wave.Furthermore,we study the interaction modes of the higher-order transformed nonlinear wave,and reveal the nature of different collision modes.Finally,we study the composite shape-changed collision mode of the higher-order transformed nonlinear wave under the combined influence of time and collision.In Chapter 4,by developing the existing Darboux transformation method,a coupled high-order nonlinear Schr?dinger equation is studied,which can describe the propagation characteristics of ultrashort optical pulses in birefringent fiber fields.On the basis of the generalized Darboux transformation,a suitable seed solution is taken to give various precise vector breather wave solutions contained in the equation.Further introducing the theory of asymptotic expansion,we obtain the rouge wave solution of the equation.In addition,the dynamic behavior of the solution is analyzed by image simulation.In Chapter 5,by developing the existing Riemann-Hilbert(RH)method,a fourcomponent nonlinear Schr?dinger equation with Lax pairs of order 5 × 5 matrices is studied for the first time.First,the spectral problem of the equation is analyzed under zero boundary conditions,based on the analytical,asymptotic and symmetric properties of the Jost function and the scattering matrix,the corresponding RH problem is successfully established.Considering the case where the scattering data are simple zero points,we solve the corresponding RH problem without reflection potential,and give the exact soliton solution of the equation.Combined with specific parameters,we analyze the dynamic behavior of one-and two-soliton solutions.In Chapter 6,we summarize the full text from the perspective of summary,and reflect and look forward to the work of this thesis.