【作者】 王璐；

【导师】 于发军；

【作者基本信息】 沈阳师范大学 ， 应用数学， 2023， 硕士

【摘要】 目前,非线性薛定谔方程的研究已经取得了许多成果,但利用达布变换和Hirota双线性方法求解高阶孤子方程依旧是热点问题。近年来,高阶非线性薛定谔方程深受关注,并且在非线性薛定谔方程中加入非局域性、PT-对称性、空间位移等条件,使非线性薛定谔方程的数学结构更加丰富,加入条件后的非线性薛定谔方程依旧满足可积性,很有研究价值。因此,我们将利用齐次平衡法、达布变换和Hirota双线性方法解决一些新型的非线性薛定谔方程的求解问题,本文主要做以下几方面的研究:第二章中,利用齐次平衡法研究时间分数阶KdV-mKdV方程。首先利用齐次平衡法求解组合KdV-mKdV方程,然后引入一致分数阶导数的定义,并结合齐次平衡法求解时间分数阶KdV-mKdV方程,得到五组新颖的精确解。第三章中,主要研究了两类连续的非线性薛定谔方程。第一类是利用达布变换研究了2+1-维非局域非线性薛定谔方程,该方程是1+1-维非局域非线性薛定谔方程的推广。首先根据给定的Lax对构造2+1-维非局域非线性薛定谔方程的达布变换,然后求解零背景下的N-孤子解的公式,进而得到2+1-维非局域非线性薛定谔方程的1-孤子解和2-孤子解。第二类是利用达布变换研究四分量耦合非线性薛定谔方程。依照已有的Lax对构造5×5谱问题,再通过达布变换,求出零背景下的1-孤子解和非零背景(q_i=e^-2it)下的1-孤子解,得到新一类的暗-亮-亮-亮孤子解,丰富了矢量孤子碰撞的研究。第四章中,主要是利用达布变换求解离散的PT-对称的非局域非线性薛定谔方程。首先根据已有的Lax对满足一定条件,找到变换矩阵T。然后利用达布变换获得离散的PT-对称的非局域非线性薛定谔方程的孤子解,分别得到在零背景和非零背景(Q_n（t）=e^2it,R_n（t）=e^-2it)下的N-孤子解形式,进而得到1-孤子解和2-孤子解,其中非零背景下的孤子解是指数形式。第五章中,主要研究两类连续的非线性薛定谔方程。第一类是研究PT-对称的2+1-维非局域非线性薛定谔方程,首先利用广田直接方法探究了方程的双线性形式,其次利用双线性方法得到PT-对称的2+1-维非局域非线性薛定谔方程的1-孤子解和2-孤子解。然后尝试改变积分常数的取值和函数的取值方式,最终得到一种新的怪波解。第二类主要是研究高阶薛定谔方程的孤子解,在原有方程的基础上引入空间位移,得到3阶非局域非线性薛定谔方程,再通过双线性算子获得方程的双线性形式,最终获得方程的1-孤子解和2-孤子解。更多还原

【Abstract】 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.更多还原