Soliton Solutions Study Of Multiclass Nonlinear Schr(?)dinger Equations

At present,many results have been achieved in the study of nonlinear Schr（?）dinger equation,however,with the Darboux transform and Hirota bilinear method approach to solving high-order soliton equations remains a hot topic.In recent years,the higher-order nonlinear Schr（?）dinger equation has received much attention,and the conditions of nonlocality,PT-symmetry,and spatial displacement are added to the nonlinear Schr（?）dinger equation to enrich the mathematical structure of the nonlinear Schr（?）dinger equation,the nonlinear Schr（?）dinger equation after adding the conditions still satisfies the integrability,which is valuable to study.Therefore,we will use the homogenous equilibrium method,Darboux transform and Hirota bilinear method to solve some new types of the nonlinear Schr（?）dinger equation solution problems,we mainly do the following research.In the second chapter,the time-fractional order KdV-mKdV equation is studied using the homogenous equilibrium method.Firstly,the homogeneous equilibrium method is used to solve the combined KdV-mKdV equation,then the definition of the uniform fractional derivative is introduced,the time-fractional KdV-mKdV equation is solved by combining the homogeneous equilibrium method to obtain five novel sets of accurate solutions.In the third chapter,two types of continuous nonlinear Schr（?）dinger equations are studied.The first type is the 2+1-dimensional nonlocal nonlinear Schr（?）dinger equation by using of the Darboux transform,which is a generalization of the1+1-dimensional nonlocal nonlinear Schr（?）dinger equation.Firstly,the Darboux transform of the 2+1-dimensional nonlocal nonlinear Schr（?）dinger equation is constructed from the given Lax pair,then the formula for the N-soliton solution in the zero seed is solved,which leads to the 1-soliton solution and the 2-soliton solution of the 2+1-dimensional nonlocal nonlinear Schr（?）dinger equation.The second type is four-component coupled nonlinear Schr（?）dinger equation using the Darboux transform.The 5×5 spectral problem is constructed according to the existing Lax pair,then through the Darboux transform obtain the 1-soliton solutions with the zero seed and the 1-soliton solutions with the non-zero seed(q_i=e^-2it),a new class of dark-light-light-light soliton solutions is obtained,which enriches the study of vector soliton collisions.In the fourth chapter,the main focus is on solving the discrete PT-symmetric nonlocal nonlinear Schr（?）dinger equation using the Darboux transform.First,the transformation matrix T is found based on the existing Lax pairs satisfying certain conditions.Then the soliton solutions of the discrete PT-symmetric nonlocal nonlinear Schr（?）dinger equation are obtained using the Darboux transform,and the N-soliton solution forms are obtained in the zero seed and the non-zero seed(Q_n（t）=e^2it,R_n（t）=e^-2it),respectively,which leads to the 1-soliton solution and the 2-soliton solution,where the soliton solution in the non-zero seed is exponential.In the fifth chapter,it mainly studies two types of continuous nonlinear Schr（?）dinger equations.The first type is the study of the PT-symmetric 2+1-dimensional nonlocal nonlinear Schr（?）dinger equation.Firstly,the bilinear form of the equation is explored using the direct method of Hirota.Secondly,the bilinear method is used to obtain the1-soliton solution and the 2-soliton solution of the PT-symmetric 2+1-dimensional nonlocal nonlinear Schr（?）dinger equation.Then an attempt is made to change the values of the integration constants and the way the functions are taken to finally obtain a new kind of strange wave solution.The second type mainly studies the soliton solutions of the higher-order Schr（?）dinger equation,by introducing spatial displacement on the basis of the original equation,obtains the 3rd-order nonlocal nonlinear Schr（?）dinger equation,then the bilinear form of the equation is obtained by the bilinear operator,and the 1-soliton solutions and 2-soliton solutions of the equation are obtained.