A novel approach to integrability of semi-discrete systems

物理现象通常用非线性微分/差分方程来描述。一般来说，大多数非线性系统的精确解析解是不可能得到的，最好的方法是使用近似、渐近或数值方法。同时，存在着一类显著的非线性系统——可积系统。可积系统可以被细致地分析，具有丰富的隐含代数结构，并且通常具有几何实现。对可积系统的兴趣正在激增。它们的应用范围和与其他数学领域的意想不到的联系正在迅速增长。因此，中心问题是确定给定方程是否可积，并最终给出可积系统的完整分类。在这个项目中，我们提议测试受最近发展启发的可积系统理论中的新方法。我们提出用差分代数的严格术语重新表述可积性问题。我们的新思路是研究相应差分域的非局部扩展，以便在形式伪差分级数的应用中获得更好的灵活性，并旨在找到全称必要可积条件。我们将开发一个涉及对象的符号表示，以对称洛朗多项式的形式重新定义问题（类似的想法在微分情况下被证明是成功的，但在差分设置中尚未开发）。我们的目标是在一个长期存在的问题上取得进展，这个问题的解决将对数学、数学物理、数值分析等领域的发展产生持久的影响。我们还将尝试将这些新方法扩展到非交换（自由结合和量子）设置，以探索非交换可积系统的未知领域，它们的泊松结构和量子化。最近出现的一种新的量化方法是基于对自由结合代数中有值函数的动力系统和该代数的某些不变微分理想的研究。我们的目标是将半离散可积系统理论扩展到自由结合和量子化代数领域，并将其与非交换代数几何、量子和统计力学联系起来。这个小的研究项目是一个跳板或可行性研究的未来全面的研究项目适合标准模式的应用。

Physical phenomena are often described in terms of nonlinear differential/difference equations.In general it is impossible to obtain exact analytic solutions to most nonlinear systems and the best one can do is to use approximate, asymptotic or numerical methods. Meanwhile, there exists a remarkable class of nonlinear systems called integrable systems. Integrable systems can be analysed meticulously, possess rich hidden algebraic structures and often have geometric realisations. The interest in integrable systems is surging. The range of their applications and unexpected connections with other areas of Mathematics is growing fast. Therefore, the central problems are to determine whether a given equation is integrable, and ultimately to make a complete classification of integrable systems. In this project we propose to test new methods in the theory of integrable systems inspired by recent developments. We propose to re-formulate the problem of integrability in rigorous terms of difference algebra. Our novel idea is to study non-local extensions of the corresponding difference field in order to achieve a better flexibility in the application of formal pseudo--difference series and aiming to find universal necessary integrability conditions. We are going to develop a symbolic representation of the objects involved to re-cast the problem in terms of symmetric Laurent polynomials (similar ideas proved to be successful in the differential case, but have not been developed in the difference setting). We aim to make a progress in a long standing problem, whose solution would have a lasting impact on the development of Mathematics, Mathematical Physics, Numerical Analysis and far beyond. We also will attempt to extend these new methods to non-commutative (free associative and quantum) settings to explore uncharted terrain of non-commutative integrable systems, their Poisson structures and quanisations. A recently emerged new approach to quantisation is based on the study of dynamical systems for functions with values in a free associative algebra and certain invariant differential ideals of the algebra. We aim to extend the theory semi-discrete integrable systems to free associative and quantised algebra domains and to link it with non-commutative algebraic geometry, quantum and statistical mechanics. This small research project is a spring-board or feasibility studies for the future full scale research projects suitable for standard mode applications.