高维非线性可积系统的子代数优化、非局域对称以及精确解

对称方法是有效的研究可积系统性质的方法之一。Lie对称算子生成的Lie群往往拥有无穷多个子群，研究群不变解的分类问题，需要探究子代数优化系统。非局域对称的无穷小当中包含有独立变量的积分，可以反映出独立变量的全局行为，有助于得到新的群不变解。本项目重点研究高维非线性可积系统的子代数优化与非局域对称，具体包括三个方面：(1)将Ibragimov子代数优化方法扩展到高维Lie代数的研究，建立一套快速判别有效群变换、精准筛选代表元的方法，给出子代数优化系统，并研究解的群分类问题；(2)利用截断Painlevé展开法构造几类高维非线性可积系统的非局域对称，研究Bäcklund变换、线性叠加非局域多残余对称、n阶Bäcklund变换以及精确解等性质；(3)建立几类规范约束，消去规范自由度对势系统的影响，将守恒律构造势系统方法推广到高维情形，研究非局域对称，并利用非局域对称寻求新解。

The symmetry method is one of the effective methods to investigate some properties of the integrable systems. Lie groups generated by Lie symmetry operators often have infinite subgroups. In order to classify the equivalent group invariant solutions, the optimal systems of subalgebras of Lie algebras need to be investigated. The integration of independent variables in the infinitesimals of nonlocal symmetries can reflect the global behavior of independent variables, which is helpful to obtain the new group-invariant solutions. This project focuses on the optimal systems of subalgebras and nonlocal symmetries of the high-dimensional nonlinear integrable systems, which includes three aspects. (1) The method constructing the optimal systems of subalgebras proposed by Ibragimov is extended to the high-dimensional Lie algebras. A method to quickly distinguish effective group transformations and accurately select representative elements is established. Some optimal systems of subalgebras are derived, and the group classifications of solutions are studied. (2) The nonlocal symmetries of several nonlinear integrable systems are constructed by means of the truncated Painlevé expansion method. The properties of Bäcklund transformations, linear superposed nonlocal multiple residual symmetries, nth Bäcklund transformations and exact solutions are studied. (3) Several kinds of gauge constraints are established to eliminate the influence of gauge degrees of freedom on the potential systems. The method of constructing potential systems with conservation laws is extended to the high-dimensional case. The nonlocal symmetries are studied and the new solutions are obtained by using the nonlocal symmetries.