【作者】 张玉娟；

【导师】 赵敦；

【作者基本信息】 兰州大学 ， 数学·应用数学， 2016， 博士

【摘要】 本文分为两部分。第一部分针对几类与孤子管理问题相关的非自治非线性数学物理方程,构造了比较一般的非等谱AKNS系统,其谱参数满足具有高阶多项式非线性项的常微分方程。第二部分将半单李代数决定的AKNS系统扩展到非半单李代数确定的AKNS系统,研究了一类AKNS可积耦合系统。在第一部分中,我们导出了可积的非自治非线性薛定谔方程,包括局部的和非局部的情形,还统一导出了PT-对称的局部和非局部非自治Gross-Pitaevskii方程。在此基础上,我们给出了非自治非线性薛定谔方程的无穷多守恒律;构造了它们的达布变换和类N-孤子解的行列式表示;建立了非局部非线性薛定谔方程和具有自旋传输扭矩的非自治海森堡铁磁链方程之间的关系。进一步,我们将这些结果应用到玻色-爱因斯坦凝聚和非线性光学中的孤子管理问题以及铁磁体的磁化动力学研究中,得到了一些新的物理结果。对PT-对称的局部和非局部非自治Gross-Pitaevskii方程,我们讨论了其反散射变换。我们发现,与局部非自治Gross-Pitaevskii方程不同,PT-对称的非局部Gross-Pitaevskii方程允许特征函数的两种不同的约化,这使得它有两种不同的反散射解。在第二部分中,我们构造了一类特殊的非半单李代数,从而引入了一类新的AKNS系统的可积耦合。进一步,我们讨论了这类新的AKNS系统的可积耦合的哈密顿结构,说明了它的刘维尔可积性。给出了这类AKNS系统可积耦合的达布变换公式,计算了这类AKNS可积耦合系统中的二次和三次可积耦合方程的一次类孤子解。并将AKNS系统的可积耦合扩展到它的双可积耦合,构造了这类AKNS双-可积耦合系统的达布变换公式。更多还原

【Abstract】 This paper includes two parts. The ?rst part focuses on several nonautonomous nonlinear mathematical physics equations about soliton management, constructs a general non-isospectral AKNS system, in which the spectral parameter is determined by an ordinary di?erential equation with high order polynomial nonlinearity. The second part generalize the AKNS hierarchy determined by the semisimple matrix Lie algebra to the AKNS hierarchy determined by a non-semisimple matrix Lie algebra,and investigate a kind of AKNS integrable couplings.In the ?rst part of this paper, we introduce integrable nonautonomous nonlinear Schr¨odinger equations, which includes the local and the nonlocal case, and obtain PT symmetric local and nonlocal nonautonomous Gross-Pitaevskii equations. Based on this, we present the in?nite conservation laws of the nonautonomous nonlinear Schr¨odinger equation; construct their Darboux transformation and present the determinant representation of the N-fold soliton-like solutions; and construct the connection between the nonlocal nonlinear Schr¨odinger equation and the nonautonomous Heisenberg spin chain equation with time-dependent inhomogeneous bilinear interaction and spin-transfer torque. Furthermore, we apply these results to the soliton management in Bose-Einstein condensates and nonlinear optics, the dynamics in ferromagnetic magnetism, and obtain some new physical results. For the PT symmetric local and nonlocal nonautonomous Gross-Pitaevskii equations, we discussed their inverse scattering transformation. We ?nd that unlike the local case,the PT-symmetry of the nonlocal Gross-Pitaevskii equation allows two di?erent choices of the symmetry reductions of the eigenfunctions which guarantee that it has two di?erent kinds of inverse scattering solutions.In the second part of this paper, we construct a kind of non-semisimple matrix Lie algebra, and introduce a novel hierarchy of integrable couplings for the AKNS system. Moreover, we discuss the bi-Hamiltonian structures of the resulted coupling system, and the Liouville integrability has been proved. We also propose a formulation of Darboux transformation for the couplings and present the exact one-soliton-like solution for the integrable couplings of the second and third order AKNS equations. In addition, we exploit AKNS integrable couplings to bi-integrable couplings of the AKNS system, and generalize the Darboux transformation to the bi-integrable couplings of the AKNS system.更多还原