The Exact Solutions Of The Generalized Mixed Nonlinear Schr?dinger Equation

With the development of soliton theory,a large amount of results on the construction of exact solutions of nonlinear partial differential equations have been accumulated.Among the many solving methods,the Riemann-Hilbert method and Darboux transformation are two very important methods.The former is an analytical method,while the latter essentially is an algebraic method.In this thesis,we will use Riemann-Hilbert method and Darboux transformation to investigate an important nonlinear mathematical and physical model,i.e.,the generalized mixed nonlinear Schodinger(GMNLS)equation.The main results are listed as following.1.The Riemann-Hilbert methods to investigate the soliton solutions of the GMNLS equation with both zero and nonzero boundary conditions at infinity are carried out,respectively.Through spectral analysis,we obtain the sectionally analytic matrixvalued functions,by which the corresponding Riemann-Hilbert problems are formulated.Solving the Riemann-Hilbert problems for the reflectionless cases,the N-soliton solutions of the GMNLS equation are recovered.To obtain the explicit forms of the solutions,the expressions of two key factors J0 and eif-σ3 are obtained,respectively.For the applications of N-soliton formula with zero boundary condition,the dynamical features of one-,two-and three-soliton solutions are discussed and the interactions of N-solitons are established.Using the N-soliton formula with nonzero boundary condition,we discuss the dynamics evolutions of one-soliton and one-breather solutions.2.An explicit Darboux transformation of the GMNLS equation is presented.The compact determinant representation of the N-fold Darboux transformation of the GMNLS equation is constructed and the Nth-order solution is built.From the vacuum solution,two different kinds of explicit one-soliton solutions are constructed and discussed.From the plane wave solution,a solution with six free parameters is constructed,and through the detailed analysis,we conclude that by choosing different values of parameters,this solution is classified as:periodic wave,dark soliton and bright soliton.3.Two equivalent ways to construct the generalized N-fold Darboux transformation of the GMNLS equation are presented.With the help of the generalized Darboux transformation,several novel rational solutions with both zero and nonzero backgrounds are formulated.The dynamics evolutions of these rational solutions are illustrated by plots.Especially,for the first-order rational solutions with nonzero background of case 2 and case 3,we find that three parameters a,b,β(related to the effects of self-steepening,self phase-modulation and quintic nonlinearity in the GMNLS equation)will induce four kinds of state transitions and five kinds of state transitions,respectively.4.The relationships between the solutions and Darboux transformations of GMNLS equation and the Kundu-Eckhaus equation are built.We prove that under the second kind of properties for the spectral parameters and eigenfunctions,and setting b=-2,a→0,the solution of the GMNLS equation will degenerate into the solution of the Kundu-Eckhaus equation,and the Darboux transformation of the GMNLS equation by multiplying a factor will also degenerate into the Darboux transformation of the Kundu-Eckhaus equation.