Loop代数上一种Toda力学系统的求解问题

Study of Solving a Toda Dynamic System with Loop Algebra

研究Loop代数上的一种Toda系统L=[L,M],其Lax Pair中的M是反对称矩阵,而L=L^++M,L^+是准上三角矩阵(包含对角部分),证明这种系统的Lax方程的求解问题与相关的正则Riemann- Hilbert问题等价．按此方法,发现在某些特定的初值条件下系统是可积的．并给出实例求解这一问题,得到了精确解．

In this paper, we construct a Toda system with Loop algebra, and prove that the Lax equation L^·= [L, M] can be solved by means of solving a regular Riemann-Hilbert problem. In our system, M in Lax pair is an antisymmetrical matrix, while L = L^＋ ＋ M, and L^＋ is a quasi-upper triangular matrix of loop algebra. In order to check our result, we exactly solve a R-H problem under a given initial condition as an example.