超对称可积系统的可积性及其解的构造

The study of supersymmetric integrable systems has drawn as much attention specially due to their potential applications in high energy physics. In the context of integrable models, it has resulted in the supersymmetrization of a number of integrable equations and the extension of the methodologies involved in the study of integrable hierarchies to the supersymmetric framework. At present, a number of important integrable systems have been embedded into their supersymmetric counterparts. However, the supersymmetric integrable systems are much more complicated than to study the integrable pure bosonic systems. This project will devoted to the following investigation for the supersymmetric integrable systems: 1. Based on the bosonization approach,the supersymmetric integrable systems can be changed to a system of coupled bosonic equations. 2. The traveling wave solutions and the similarity reduction solutions of the model are obtained with the mapping and deformation method and the Lie point symmetry theory, respectively. 3. By applying the fermionization approach, the inverse version of the bosonization approach, the new supersymmetric integrable extensions can be found from known integrable boson systems and Painlevé analysis. Furthermore, the integrable property and the exact solutions to the supersymmetric extensions will be studied. The work will be helpful to reveal essential properties of the supersymmetric integrable systems. It can also provide the theoretical foundation for the possible physical applications of these supersymmetric integrable systems.

由于超对称可积模型在高能物理中的应用，超对称可积系统的研究引起人们广泛的关注。在超可积系统领域中，经典可积系统的超对称化及其超对称系统的可积性质是该领域主要的研究内容。目前，已经构造了大量具有潜在应用价值的超对称可积系统。然而，超可积系统中费米场的存在，给研究带来许多困难。基于费米场引起的困惑，本项目将对超对称可积模型展开以下研究：1) 利用玻色化方法，将超对称可积系统，特别是复杂的N>1 超对称系统转化为只有玻色场的耦合系统；2) 结合形变映射和对称约化方法研究玻色化后的耦合系统，试图寻找超对称可积系统的行波解和对称约化解；3) 应用玻色化的逆问题，即费米化方法，结合Painlevé分析，通过普通的玻色系统构造新的超对称可积系统，并对新的超对称可积系统的精确解和可积性质做进一步研究。通过本项目的研究，有助于揭示一类超对称可积系统的基本性质，为超对称模型的实际应用提供理论。

应用玻色化方法，将超对称可积系统变为只有玻色场的非线性可积系统. 该方法可以避免费米场反对易带来的运算困难. 结合形变映射方法、李点对称约化方法和非局域对称约化等方法构造了超对称可积系统的精确解等可积性质. 同时，将玻色化方法推广到N=2超对称系统中，构造了N=2超对称对称约化解等其他类型的精确解. 拓展相容tanh函数方法，将相容tanh函数方法应用于不可积的非线性系统，具体研究了CGKP、(3+1)-KP和(2+1)-Boussinesq方程，得到了这些方程的孤子与其他周期波相互作用解，拓展了相容tanh函数方法的适用范围. 利用Painlevé截断方法得到了一类可积系统的非局域对称，为了求解非局域对称对应的初始问题，通过合理的拓展原可积系统，将非局域对称局域对称变为点对称. 再利用李点对称方法约化拓展后的系统，得到了非线性可积系统的相互作用解，丰富了非线性可积系统解的类型.