On the algebraic structure of rational discrete dynamical systems

二维离散算子和代数曲线上的有理函数

Two-dimensional discrete operators and rational functions on algebraic curves

本文研究二维离散算子与有限间隙薛定谔算子的联系，为新类二维可积离散算子找到谱数据，并证明亏格一情形下薛定谔算子可作为离散算子极限。

Special Functions of the Isomonodromy Type, Rational Transformations of Spectral Parameter, and Algebraic Solutions of the Sixth Painlevé Equation

We discuss relations which exist between analytic functions belonging to the recently introduced class of special functions of the isomonodromy type (SFITs). These relations can be obtained by application of some simple transformations to auxiliary ODEs with respect to a spectral parameter which associated with each SFIT. We consider two applications of rational transformations of the spectral parameter in the theory of SFITs. One of the most striking applications which is considered here is an explicit construction of algebraic solutions of the sixth Painlevé equation.

Rational solutions for three semi-discrete modified Korteweg-de Vries type equations

In this paper, we consider three semi-discrete modified Korteweg-de Vries type equations which are the nonlinear lumped self-dual network equation,the semi-discrete lattice potential modified Korteweg-de Vries equation and a semi-discrete modified Korteweg-de Vries equation. We derive several kinds of exact solutions, in particular rational solutions, in terms of the Casorati determinant for these three equations respectively. For some rational solutions, we present the related asymptotic analysis to understand their dynamics better.

Rational Solutions for the Discrete Painlevé II Equation

The rational solutions for the discrete Painlevé II equation are constructed based on the bilinear formalism. It is shown that they are expressed by the determinant whose entries are given by the Laguerre polynomials. Continuous limit to the Devisme polynomial representation of the rational solutions for the Painlevé II equation is also discussed.

代数孤子在质量Thirring模型中的解

Rational solutions for algebraic solitons in the massive Thirring model

本文研究了质量Thirring模型的代数孤子解，通过双Wronskian行列式构造了理性解的层次结构，并证明了每个解由一个次数为N²的多项式定义，具有特定的极点分布。

Using discrete Darboux polynomials to detect and determine preserved measures and integrals of rational maps

In this Letter we propose a systematic approach for detecting and calculating preserved measures and integrals of a rational map. The approach is based on the use of cofactors and Discrete Darboux Polynomials and relies on the use of symbolic algebra tools. Given sufficient computing power, all rational preserved integrals can be found. We show, in two examples, how to use this method to detect and determine preserved measures and integrals of the considered rational maps.