标题： WKB分析关于修正仿射Toda场方程的线性问题 显示英文标题

标题： WKB analysis of the linear problem for modified affine Toda field equations

摘要： 我们研究了修正仿射Toda场方程的线性问题解的WKB分析，这等价于在ODE/IM对应中研究的高阶常微分方程(ODE)。 经过规范变换后，我们将线性问题的平直连接对角化，从而将后者简化为一组独立的一阶线性微分方程。 我们对低秩的经典仿射李代数显式地执行此过程。 特别是，我们研究了$D_r^{(1)}$和$D^{(2)}_{r+1}$型线性问题的WKB解，这些对应于带有伪微分算子的高阶ODE。 对角化的连接来自伴随线性问题的Riccati方程，并且与Drinfeld和Sokolov构造的可积层次中的守恒流有关，最多相差全导数。 显示英文摘要

摘要： We study the WKB analysis of the solutions to the linear problem for a modified affine Toda field equation, which is equivalent to the higher-order ordinary differential equation (ODE) studied in the ODE/IM correspondence. After gauge transformation, we diagonalize the flat connection of the linear problem to reduce the latter to a set of independent first-order linear differential equations. We explicitly perform this procedure for classical affine Lie algebras with lower ranks. In particular, we study the WKB solutions of the $D_r^{(1)}$- and $D^{(2)}_{r+1}$-type linear problems, which correspond to the higher-order ODEs with the pseudo-differential operator. The diagonalized connection is obtained from the Riccati equations of the adjoint linear problem and related to the conserved currents of the integrable hierarchy constructed by Drinfeld and Sokolov up to total derivatives.