有限变换、非局域对称及非线性波的相互作用解

The symmetry method is one of the most effective and important method in nonlinear science. The point symmetry and general symmetry are familiar to people, while it is much difficult to find and apply the nonlocal symmetry of nonlinear system. This project will be arranged as follows: Firstly, starting from a variety of known classic finite transformations (including Darboux transformation, Backlund transformation, bilinear and multilinear transformation, conformal transformation, etc.), we will reveal that its corresponding symmetry algebra is just the nonlocal symmetry algebra. Secondly, combining the obtained nonlocal symmetry with the finite transformation, some integrable problems of nonlinear system are investigated, for instance, localizing nonlocal symmetry to derive the Schwarzian form of discrete system, combining nonlocal symmetry with the nonlinearization method to construct some new finite-dimensional integrable models, making reductions via nonlocal symmetry to find the interaction between the solitary wave and other nonlinear waves, and so on. Finally, we will combine and improve some classic integrable method (inlcuding the truncated Painleve analysis method , the separation of variables method , etc.) to study the interaction between various nonlinear excitations to reveal the new phenomena in physics.

对称方法是研究非线性科学的最有效和最重要的方法之一，其中从点对称到一般对称是被大家所熟知的，而对于非局域对称，无论是其获得还是应用，长久以来就是一个难题。本项目将首先从各种已知的经典有限变换（包括达布变换、贝克隆变换、双线性和多线性变换、共形变换等）出发，揭示其对应的对称代数就是非局域对称代数。其次，结合所得非局域对称与相应有限变换，研究非线性系统的一些可积性问题，例如：通过非局域对称局域化给出离散系统Schwarzian形式；利用非局域对称做非线性化问题，构造新的有限维可积模型；进行非局域对称对应的约化求解，揭示孤立波和其他非线性波之间的相互作用；等等。最后，我们将组合和改进各种经典的可积方法(例如潘勒韦截断分析法、分离变量法等)，用来研究各种非线性激发之间的相互作用,以揭示物理学中的新现象。

本项目主要研究了有限变换、非局域对称及非线性波的相互作用解，针对项目中提到的四项研究对象，基本内容已经完成，具体为：从双线性形式的贝克隆变换和其他有限变换出发，得到了其对应的非局域对称；将非局域对称局域化，给出了相应的李代数、有限群变换和其Schwarzian 形式；通过非线性化方法构造了负梯队；得到了孤立波和其他非线性波的相互作用解。 研究成果以三篇SCI论文的形式给出，其中一篇发表在Chinese Physics B，正在出版中；另外两篇在投Journal of Mathematical Physics和Studies in Applied Mathematics。