超可积和非局域可积系统的构造及相关研究

鉴于超对称可积模型在矩阵模型的超拓展、非微扰二维超引力和拓扑场理论中潜在的物理应用，超对称可积系统及相关研究已引起人们的极大关注，此外，Ablowitz和Musslimani构造了具有宇称-时间对称势的非局域非线性Schrödinger方程，该方程在光学和光子晶体等领域均有很好的应用。目前，超可积系统以及非局域可积系统的研究已成为孤立子理论的研究热点。本项目拟对超可积系统和非局域可积系统开展以下研究：1.应用可积系统的Lax对，计算其费米场形式的非局域对称，从而构造新的超可积系统；2.利用超可积系统被玻色化后的系统，用可积系统的Lax对等方法构造新的非局域可积系统；3.利用新可积系统的非局域对称，研究新可积系统的相互作用解。本项目的实施，将有助于丰富非线性可积系统的类型，拓展可积系统理论中对称性研究方法的应用范围，揭示非线性系统的性质及相互联系，为非线性可积模型的物理应用提供理论参考。

Supersymmetric integrable systems have attracted an increasing amount of interest, mainly due to their potential physical applications in super-extensions of matrix models, non-perturbative 2D super-gravity and topological field theories. Besides, Ablowitz and Musslimani proposed a nonlocal nonlinear Schorödinger equation with a parity-time (PT) symmetric potential. The fields of optics and photonics have provided a test bed for the experimental observation of the unique phenomena related to PT symmetry. Recently, the study on super integrable and nonlocal integrable systems has become a research focus in soliton theory. .This project is devoted to the following investigations for some supersymmetric integrable and nonlocal integrable systems. 1. The nonlocal symmetry of the fermionic field is obtained by means of the Lax pair of the integrable equation. The new super integrable equation can thus be constructed by virtue of the symmetry in fermionic form. 2. The nonlocal integrable systems are constructed by using the Lax pair and bosonic systems of super forms. 3. Some interactions solutions for new integrable systems are studied by the aid of their nonlocal symmetries. .The project will be helpful to enrich types for the nonlinear integrable systems, to expand areas of application on symmetry method in integrable system, and to reveal some essential properties as well as internal relations between them for the nonlinear physics systems. It can also provide us with valuable references in theory for the possible physical applications of these integrable models.