Generalization, Discretization and Application of Nonlinear Integrable Systems Associated with Toroidal Lie Algebra Symmetry

1.得到了与修正的环形李代数sl^-t；tor&gt；_2相关的离散方程的一阶修正双线性形式。该方法适用于多分量情形，并构造了SU(N)自对偶杨-Mills方程双线性形式的一半的离散化。2.以四维拉普拉斯的格林函数作为超对称行列式解的分量，重构了SU(N)自对偶杨-Mills方程的瞬子解。瞬子的位置和振幅对应于孤子的相位常数和波数，并出现在格林函数的参数中。3.应用SU(N)自对偶Yang-Mills情形的方法，导出了与2环对称性有关的孤子方程的可积离散。通过在全无限一维Toda晶格通解的Hankel行列式表达式中引入无穷多个自变量，证明了用Schur多项式表示的线性微分算子简单地应用于种子函数即可得到分量.分量母函数由两个带离散变量移位的Τ函数之比给出。

1.The bilinear form of 1st modified hierarchy is obtained for the discrete equations associated with a modification of toroidal Lie algebra sl^<tor>_2. The method is applicable to multi-component case and a discretization of half of bilinear form for the SU(N) self-dual Yang-Mills equation is constructed.2.The instanton solutions for SU(N) self-dual Yang-Mills equation are reconstructed by taking the Green functions for 4-dimensional Laplacian as the components in the persymmetric determinant solution. The positions and amplitudes of instantons correspond to the phase constants and wave numbers of solitons and appear in the parameters of Green functions.3.The integrable discretization of soliton equations associated with the 2-toroidal symmetry is derived by applying the method of the SU(N) self-dual Yang-Mills case. It is also shown that the hierarchy of equations coincides with the one obtained through the non-isospectral deformation technique.4.By introducing infinitely many independent variables in the Hankel determinant expression of general solution for full-infinite 1-dimensional Toda lattice, it is shown that the components are simply obtained by applying the linear differential operators expressed in terms of the Schur polynomial to the seed function. The generating functions of components are given by the ratio of two Τ functions with a shift of discrete variable.