Research On Symmetry,Inverse Scattering Transform And Analytical Solutions Of Some Nonlinear Differential Equations

In this thesis,we will mainly study the Lie symmetry,inverse scattering transform,conservation laws,exact solutions and soliton solutions of some nonlinear Schr¨odinger equations based on several different methods.Nonlinear differential equations can describe nonlinear phenomena in many fields,such as mathematics,biology,physics,and even finance.Therefore,the study of these equations has potential value.The study of the symmetry,inverse scattering transform and analytical solutions of nonlinear differential equations can help explain some of the corresponding physical phenomena and Engineering applications.For example,the generalized derivative higher order nonlinear Schr¨odinger equation and the chiral nonlinear Schr¨odinger's equation in(2+1)-dimensions,which describes the pulses propagation in optical fibers and the envelope of amplitude in many physical media,respectively.The structure of this thesis is as follows:In Chapter 1,we briefly introduce the research background and significance of this direction,in which the development history of conservation law and Riemann-Hilbert method are described in detail.Finally,we briefly introduce the main research contents of this thesis.In Chapter 2,the symmetric operators and commutators of the generalized derivative higher order nonlinear Schr¨odinger equation and the chiral nonlinear Schr¨odinger's equation in(2+1)-dimensions are studied by the Lie symmetry method.Then,the symmetry reduction and group invariant solutions of the equation are obtained by using the optimal system method.The explicit power series solution is found successfully on the basis of the corresponding convergence analysis.Simultaneously,through the new conservation law theory proposed by Ibragimov,we can get the conservation law of the corresponding equation.Finally,the exact traveling wave solution of the equation is obtained by the corresponding symbolic calculations method.In Chapter 3,the Riemann-Hilbert method is extended to the three-coupled fourthorder nonlinear Schr¨odinger equations for the first time,and the corresponding soliton solution is obtained.The Riemann-Hilbert problem of the equation is successfully established by combining the analytical properties of the eigenfunctions and spectral functions under spectral analysis of Lax pair.In without the reflection case,we obtain the soliton solution of the Riemann-Hilbert problem,and then obtain the soliton solution of the equation.We obtain the soliton solution of this Riemann-Hilbert problem,and then obtain the multiple soliton solutions of the equation.In addition,we give the localized structures and dynamic behaviors of one soliton solutions and two soliton solutions of the equations are displayed by selecting the appropriate parameters.In Chapter 4,the nonzero boundary problem of the nonlinear Schr¨odinger equation in laboratory coordinates is studied for the first time,and some soliton solutions are given.By analyzed the asymptotic Lax pairs,the Jost function,scattering matrix and their analyticity and symmetries are successfully obtained.The asymptotic analysis,trace formulae and “” conditions of discrete points are obtained for the first time.By solving the Riemann-Hilbert problem,some soliton solutions of the equation are obtained.Finally,we also extended its to the double poles,and establish the corresponding discrete spectrum,residual conditions,trace formulae and “” conditions.Some features of these soliton solutions caused by the influences of each parameters are analyzed graphically in order to control such nonlinear phenomena.In Chapter 5,we studied the bright dark soliton solutions of a generalized Hirota equation with the zero-order dissipation,the generalized nonlinear Schr¨odinger equation and the two-dimensional complex Ginzburg-Landau equation based on the amplitude ansatz method.We studied the stability of the equation for the first time,and the modulation instability analysis of the equation is also analyzed by the standard linear stability analysis method.The traveling wave solution and Gaussian soliton are also givened.In Chapter 6,We used the binary Bell polynomial theory to construct a bilinear form of a(3+1)-dimensional non-integrable KdV-type equation and the simplified(3+1)-dimensional B-type Kadomtsev-Petviashvili equation,and the corresponding soliton solution is further deduced.By using the extended homoclinic text method,the homoclinic respiratory wave solution of the square path is obtained for the first time,and the rogue wave solution is further deduced.Then,we get the lump solution of the equation.It is also extended to(3+1)-dimensional gKP equation and(3+1)-dimensional vcgBKP equation,and the corresponding lump solution is obtained.Finally,the lumpoff solution and instanton/rogue wave solution of the equation are derived.In the last chapter,we briefly summarize and prospects to this thesis.