A Miura of the Painlevé I Equation and Its Discrete Analogs

We present Miura transformations for the continuous and several discrete Painlev\'e I equations. In the case of the continuous P_I, we use the Hamiltonian formulation of the Painlev\'e equations and show that there exists a Miura transformation between P_I and the binomial, second degree, equation of Cosgrove SD_V. In the case of the discrete P_I's we obtain two different kinds of Miuras. One kind relates a d-P_I to some other d-P_I while the other leads to discrete four-point equations which are the discrete analogs of the derivative of Cosgrove's equation SD_V.     

Separation of Variables for a Lattice Integrable System and the Inverse Problem

We investigate the relation between the local variables of a discrete integrable lattice system and the corresponding separation variables, derived from the associated spectral curve. In particular, we have shown how the inverse transformation from the separation variables to the discrete lattice variables may be factorized as a sequence of canonical transformations.     

子午面大视场、孤矢面大孔径的耦合系统设计

本文提出了一套由４块柱面透镜组成的激光二极管阵列泵浦固体激光器的耦合系统，它可以将尺寸为４０ｍｍ×３．６ｍｍ的半导体激光二极管阵列的发光面所发出的子午面发散角±１０°，弧矢面发散角为±３０°的泵浦光耦合到接收面的系统可用于半导体激光二极管阵列泵浦固体激光器的研究。