AKNS系统的推广及应用

发布时间：2018-04-26 23:24

本文选题：非等谱AKNS系统 + 达布变换　；参考：《兰州大学》2016年博士论文

【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。
[Abstract]:The first part is divided into two parts. In the first part, a general non spectral AKNS system is constructed for several non autonomous nonlinear mathematical physical equations related to the soliton management problem. The spectral parameters satisfy the ordinary differential equation with high order polynomial nonlinear term. The second part extends the semi single lie algebra to the non semi single lie. In the AKNS system determined by algebra, we study a class of AKNS integrable coupling systems. In the first part, we derive the integrable nonautonomous nonlinear Schrodinger equation, including the local and non local cases, and derive the local and non local non autonomous Gross-Pitaevskii equations of PT- symmetry. On this basis, we give non autonomy. The infinitely many conservation laws of the nonlinear Schrodinger equation and the determinant representation of their Da cloth transformation and the N- soliton like solution are constructed, and the relationship between the nonlocal nonlinear Schrodinger equation and the non autonomous Heisenberg ferromagnetic chain equation with the spin transmission torque is established. Some new physical results are obtained in the study of the soliton management in nonlinear optics and the magnetization dynamics of ferromagnets. We discuss the inverse scattering transformation for the local and non local nonautonomous Gross-Pitaevskii equations of PT- symmetry. We find that the non local PT- symmetry is different from the local nonautonomous Gross-Pitaevskii equation. The Gross-Pitaevskii equation allows two different reductions of characteristic functions, which makes it have two different anti scattering solutions. In the second part, we construct a special class of non semi single Lie algebras, and then introduce the integrable coupling of a new class of AKNS systems. Further, we discuss the integrable coupling of this new kind of AKNS system. The Liouville integrability of the AKNS system is explained. The integrable coupling transformation formula of this kind of AKNS system is given. The one class soliton solution of the two and three integrable coupling equations in this kind of AKNS integrable coupled system is calculated. The integrable coupling of the AKNS system is extended to its double integrable coupling, and this kind of AKNS double integrable coupling is constructed. The Da cloth transformation formula of the system.

【学位授予单位】：兰州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.29

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