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

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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.