Quantization of difference nonlinear equation of Painleve type

Panleve方程是20世纪初发现的一类具有良好性质的二阶非线性常微分方程。它们是一类具有仿射Weyl群对称性的非自治Hamilton系统，通过降维从单值保持方程和孤子偏微分方程中产生。Painleve型方程是那些具有高对称性的方程，通常是部分或高阶的，以类似的方式产生。本文研究Painleve型方程及其差分形式的正则量子化。特别地，我们通过量子化孤子系统的约化给出了量子化差分Painleve方程，以及作为共形场论推广的公式。

Panleve equations are second order nonlinear ordinary differential equations with certain good properties discovered in early 1900s. They are nonautonomous Hamiltonian systems allowing affine Weyl group symmetry, arise from monodromy preserving equations as well as solitonic partial differential equations through dimensional reduction. Painleve type equations are those equations with high symmetry, partial or of higher order in general, arising in similar ways. We investigate the canonical quantization of Painleve type equations and difference versions thereof. In particular we gave the quantized difference Painleve equations via the reduction of quantized solitonic systems, as well as the formulation as generalization of conformal field theories.