超对称可积方程的构造及其在数学物理中的应用

本项目的研究对象为超对称可积系统。超对称海森堡铁磁链模型是一个重要的超对称可积模型，利用引入辅助自旋矩阵的方法，我们将研究 (2+1) 维ML-I方程、M-II方程以及 M-III方程推广到超对称的情形，从而构造高维推广的超对称海森堡铁磁链模型并研究其可积性质；我们还将探讨推广的Hirota方程的超对称化，通过规范变换，建立超对称推广的Hirota方程与推广的超对称海森堡铁磁链模型之间的联系，并对它们的解进行必要的分析和研究；通过构造恰当的傅里叶变换场以及超对称的哈密顿量，我们将建立超的无穷维 3-代数与超的非色散KdV 族之间的联系。

This project is devoted to investigate the supersymmetric integrable systems . The Heisenberg supermagnet model is an important supersymmetric integrable system. By introducing the auxiliary matrix variable, we shall construct the supersymmetric extension of ML-I, M-II and M-III equation and study the integrability of the system. We will construct the superextensions of the generalized Hirota equation with linear inhomogeneities , and we try to derive the corresponding equivalent generalized Heisenberg supermagnet model in terms of the gauge equivalent. The solution of the systems will be analyzed. By means of the construction of the suitable Fourier transform field and the supersymmetric Hamiltonian, we try to establish the relation between the infinite-dimensional 3-algebra and the super dispersionless KdV hierarchy.