多分量（超对称）可积系统的构造以及离散可积系统中若干问题的研究

本项目主要研究超对称和离散可积系统的构造、性质和应用。拟从以下几方面展开研究：.①利用在交换子代数中取值的方法，将（1+1）维和（2+1）维推广的海森堡铁磁链模型推广到多分量形式，并研究其Lax表示，几何等价方程, Bäcklund变换以及几何等价方程的非局域对称。②构造多分量的（1+1）维和（2+1）维推广的超对称海森堡铁磁链模型，并研究相应的Lax表示，规范等价方程及其Bäcklund变换等可积性质。③推广的非线性薛定谔方程在光纤通讯中光孤子的研究方面具有重要应用，本项目将利用几何等价性，建立其与推广的海森堡铁磁链模型之间的关系，并研究推广的海森堡铁磁链模型的自旋波解。④研究约束离散KP系列，约束q形变KP系列在规范变换下的应用。本项目的研究，将有助于理解超对称和离散可积系统的结构和性质，为超对称和离散可积模型在数学物理中的应用提供理论参考。

This project is devoted to the following investigations for the construction，properties and application of the supersymmetric and discrete integrable systems：.①By taking values in a commutative algebra, we shall construct multi-component generalized Heisenberg ferromagnet models in (1+1) and (2+1)-dimensions. Then we will derive their Lax representation, geometrical equivalence, Bäcklund transformation and nonlocal symmetry of the integrable systems..②We try to construct multi-component generalized Heisenberg supermagnet models in (1+1) and (2+1)-dimensions and the corresponding integrable properties, such as Lax pair, gauge equivalent equations and Bäcklund transformation..③The generalized nonlinear Schrödinger equation plays a critical role in the study of optical solitons for optical fiber communications. By means of the geometrical equivalence, we try to establish the generalized Heisenberg ferromagnet models. Then we shall study their spin wave solution. .④ We plan to investigate the applications of the gauge transformation of the constraint discrete KP and q-KP hierarchies.This investigation will be helpful to understand the structures and properties of supersymmetric and discrete integrable systems and can provide a theoretical foundation for the physical applications.