非线性可积系统的非局域对称及其应用

It is well known that the Lie point method is effective to study the integrability properties and exact solutions of the nonlinear integrable equations. With the development of the symmetry reduction, the Lie point symmetry method has the limitation in studying the nonlinear integrable equations. To obtain new solutions of the nonlinear integrable systems that are not obtained through the Lie point symmetry methods, several extensions to the local symmetry approach have been proposed. A generalized symmetry depends on integrals of the dependent variables which is called the nonlocal symmetry is much less known. This project will devoted to the following investigation for the nonlocal symmetries and their applications: 1. Based on the truncated Painlevé method, the interactions between multi-soliton solutions and other nonlinear waves are constructed with introducing n-th residual symmetry. 2. The new integrable systems can be constructed by means of an asymptotically exact reduction method based on Fourier expansion. The nonlocal symmetries and interactions solutions for new systems are studied via the help of the Lax pair. 3. The nonlocal symmetries and interactions solutions are constructed with the Bäcklund transformations. 4. Infinitely many nonlocal symmetries and infinitely many nonlocal conservation laws are constructed in terms of above three methods. The work will be helpful to reveal the integrable properties and new interactions solutions of the nonlinear systems. It can also provide the theoretical foundation for the possible physical applications of these integrable systems.

李点对称方法是研究非线性系统可积性质的有效工具. 随着可积系统的深入研究，李点对称约化方法存在诸多局限性. 为了构造可积系统更多的对称和精确解，将李点对称进行拓展，即无穷小变量包含非局域变量，称为非局域对称. 本项目就非局域对称及应用做以下研究：1.应用Painlevé截断，引入N次留数对称，构造可积系统，特别是PT对称，CPT对称可积系统及超对称可积系统的多孤子解与其他孤立波相互作用解；2.通过渐进Fourier展开方法构造新的Lax意义下可积的系统，结合Lax对研究系统的非局域对称及相互作用解；3.应用可积系统的Bäcklund变换构造非局域对称及其相互作用解，研究Painlevé截断、Lax对和Bäcklund变换三种方法构造的非局域对称的关系；4.构造上述三种方法对应的无穷多非局域对称和守恒律. 通过本项目的研究，有助于了解一类非线性系统的可积性质，为解释新的物理现象提供理论参考.

应用Painlevé截断，构造了一类非线性可积系统，包括超对称可积系统的非局域对称，利用非局域对称局域化方法，将非局域对称转化为局域对称，利用对称约化方法构造了孤子与其他孤立波相互作用解. Lump解作为一种在空间维度局域的有理函数解，lump解的研究得到了广泛的关注. 应该Hirota双线性方法，构造了一类具有lump解的非线性偏微分方程. 利用ansätz方法，得到了非线性偏微分方程的lump解，lump解及其与其他类型相互作用解. 另外，在经典可积系统基础上增加了高阶非线性和色散作用效应，利用速度共振条件，可以将可积系统的多孤子解转化为孤子分子，孤子分子与孤子，孤子分子与lump等相互作用解. 通过该项目的研究，可以提供了更多有潜在物理应用的非线性系统，同时，丰富了非线性局域激发的类型.