直接线性化与离散可积系统的Lie代数分类

The study of discrete integrable systems has been recently becoming a prominent subject in the theory of integrable systems, due to the richness of their mathematical structure, and has simultaneously made a great contribution to the development of modern mathematical physics...At the current stage, the theory of discrete integrable systems is not yet complete, and there are still lots of integrable discrete models to be discovered; while considering the classification problem of discrete equations is an effective path to the success...The project is concerned with the classification of discrete integrable systems in terms of Lie algebras and relevant problems, by the so-called direct linearisation method. The concrete research topics are as follows:..(1).The direct linearisation will be connected with Lie algebras, for the purpose of giving a full classification of integrable discrete equations, and searching for novel integrable discrete models and integrability characteristics...(2).Master symmetries and generating partial differential equations for the resulting discrete models in the classification will be constructed, which will further unveil the integrable structure of discrete systems, and the interlinks between discrete equations and their corresponding continuous hierarchies...(3).The direct linearising transform for each integrable discrete equation in the classification will also be constructed, in order to explore the most general solution space in the direct linearisation scheme...The main purpose of this project is to understand discrete integrability from the aspect of the direct linearisation approach. The ultimate goal is to develop a unified framework for the study of discrete integrable systems.

离散可积系统蕴含丰富的数学结构，其研究在近年来成为了可积系统理论中的一个核心分支，同时也推动了现代数学物理的发展。现阶段，离散可积系统理论尚不完整，仍有许多未知的可积离散模型有待被发现，而考虑离散方程的分类正是解决此问题的有效途径。本项目计划探讨基于直接线性化方法的离散可积系统Lie代数分类与相关问题。研究内容包括：(1) 建立直接线性化与Lie代数的联系，给出可积离散方程的代数分类，并寻找全新的可积离散模型与可积特征；(2) 构造分类所得离散方程的主对称与生成偏微分方程，进一步揭示离散系统的可积性及其与连续方程族之间的内在联系；(3) 对分类中的可积离散方程构造直接线性变换，探索直接线性化意义下离散方程最一般的解空间。本项目旨在尝试从直接线性化角度理解全部离散可积性，最终目标是发展一个研究离散可积系统的统一框架。

离散可积系统蕴含丰富的数学结构，其研究在近年来成为了可积系统理论中的一个核心分支，同时也推动了现代数学物理的发展。现阶段，离散可积系统理论尚不完整，仍有许多未知的可积离散模型有待被发现，而考虑离散方程的分类正是解决此问题的有效途径。..本项目探讨了基于直接线性化方法的离散可积系统Lie代数分类与相关问题。研究成果包括：(1) 建立了直接线性化中线性积分方程与若干类无穷维Lie代数的联系，并从中构造了大量新的离散模型与可积特征；(2) 构造了2+2维的非自治微分-差分AKP、BKP以及CKP方程，这些方程的扮演了离散AKP、BKP与CKP方程的主对称；(3) 以2维Toda型方程为例，我们从直接线性化中诱导了关于一般有限极点解的Cauchy矩阵表示公式，从而将原有Cauchy矩阵的结果推广至BCD型代数所对应的可积系统；(4) 建立了由Date、Jimbo与Miwa所给出的一个椭圆曲线参数化的KP型方程的直接线性化格式，从中我们进一步探讨了该方程的可积性。

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