非线性离散可积方程与离散Painlevé方程族的连续极限理论

The research project seeks to establish the continuous limits theory of the nonlinear discrete integrable equation and discrete Painlevé equation hierarchy. By using the Darboux transformation, we construct the new explicit solutions for the discrete combined integrable equation (discrete Gardner equation, discrete Hirota equation and discrete Maxwell-Bloch equation), and then investigate their continuous limits yield the corresponding results for the continuous equations. We also discuss the continuous limits theory of the Lax pairs,the infinite conservation laws,the symmetries, the Hamiltonian structures , the Darboux-B?cklund transformations and exact solutions for some discrete integrable equation associated with high-order spectral problems (discrete Sawada-Kotera equation，discrete Boussinesq equation, discrte Kaup-Kupershmidt equation) and discrete multi-component integrable systems (discrete multi-component Schr?dinger system and discrete multi-component mKdV system). With the help of the continuous limits theory of the non-isospectral discrete equation hierarchy (such as non-isospectral Volterra equation hierarchy, non-isospectral discrete mKdV equation hierarchy, non-isospectral discrete Boussinesq equation hierarchy), the one-to-one relations between discrete Painlevé equation hierarchy and Painlevé equation hierarchy are established. Finally, we focus on the continuous limits theory of the properties associated to the discrete Painlevé equation hierarchy.

本项目拟研究非线性离散可积方程和离散Painlevé方程族的连续极限理论。主要内容包括利用Darboux变换构造离散组合可积方程的各种精确解，并建立这些解与对应连续系统解的一一对应关系。研究与高阶谱问题相联系的离散可积方程(离散Sawada-Kotera方程，离散 Boussinesq 方程, 离散Kaup-Kupershmidt方程，离散多分量Schr?dinger 系统)的Lax对，无穷守恒率，哈密尔顿结构，对称，Darboux-B?cklund变换及精确解的连续极限理论。通过构造非均匀谱Volterra方程族，非均匀谱离散mKdV方程族，非均匀谱离散Boussinesq 方程族的连续极限理论，建立离散Painlevé (I, II, III)方程族与Painlevé (I, II, III)方程族之间的一一对应关系，进一步研究离散Painlevé方程及其族的各种性质的连续极限理论。

非线性可积偏微分方程的离散化是非线性动力系统领域的一个前沿问题。基于离散可积模型在数值模拟和离散物理系统中的广泛应用，人们希望所构造出的离散方程尽可能多的保持各种可积性质。本项目研究的是在此基础上的深化问题：如何建立离散可积系统精确解及可积性质与相对应连续模型之间的一一对应关系？我们主要利用离散可积系统的连续极限理论，即在一个统一的极限框架下，严格证明由离散可积模型的精确解和各种性质确实能够一一对应于连续相对应的结果。在本项目的研究中，我们提出一系列新的空间离散可积模型，例如：多分量局部耦合非线性薛定谔系统，Hirota方程，Burgers和Sharma-Tasso-Olver组合方程，Kundu-Echkaus 方程。进一步成功构造这些离散方程的Lax对，无穷多个守恒率，Darboux变换等可积性质。最后引入Darboux 变换的连续极限理论并借助于Lax对的连续极限理论实现了各种精确解之间的一一对应关系；通过线性组合技巧建立无穷多个守恒率的连续极限理论。