Computer algebra and the integrability of continuous and discrete systems

在数学和物理学中，孤子是一种自我增强的孤立波，由介质中非线性和色散效应之间的微妙平衡引起。物理学中出现孤子的现象包括：特殊的水波、大气波和非线性光学中的波。在工业上，通过光纤系统传输孤子可以提高光通信的性能。具有孤子解的偏微分方程（PDEs）表现出许多特征性质，如所谓的贝克伦变换的存在，这种变换可以从简单的孤子解产生更复杂的孤子解，以及解的非线性叠加原理的存在。相对年轻的离散微分几何领域能够从一个更深层次的原理推导出这些偏微分方程的所有性质：由满足所谓面关系的n维离散面构建的离散（n+1）维网格的一致性。从这些（n+1）维网格中，可以通过适当的限制程序推导出具有孤子解的偏微分方程及其贝克伦变换和叠加原理。本提案的目的之一是研究这种面关系（例如，确定3维和4维面关系的主要类别），并为此目的开发计算机代数算法和改进解决涉及10^17项的空前规模的相关代数条件所需的技术（这些技术随后将有其他应用，并且对商业计算机代数系统也有潜在的兴趣）；因此，提出一个关于四维可积孤子系统是否存在的命题，从而有助于对偏微分方程的可积性有更深的理解。这项研究将与来自柏林工业大学和慕尼黑大学的该领域领先的国际专家合作完成。

In mathematics and physics a soliton is a self-reinforcing solitary wave caused by a delicate balance between nonlinear and dispersive effects in the medium. Phenomena in physics where solitons occur include: special water waves, atmospheric waves, and waves in non-linear optics. Solitons are industrially relevant in transmission through fiber optical systems that increases the performance of optical telecommunications. Partial differential equations (PDEs) that have soliton solutions show many characteristic properties, as the existence of so-called Baecklund transformations that generate more complicated soliton solutions from simpler ones and the existence of non-linear superposition principles of solutions. The relatively young field of Discrete Differential Geometry is able to derive all these properties of such PDEs from a deeper principle: the consistency of a discrete (n+1)- dimensional grid that is built from n - dimensional discrete faces that satisfy a so-called face relation. PDEs with soliton solutions, together with their Baecklund transformations and superposition principles can be derived through suitable limiting procedures from those (n+1) - dimensional grids. One aim of this proposal is to study such face relations (e.g. the determination of major classes of 3- and 4-dimensional face relations) and for that purpose to develop computer algebra algorithms and refine techniques that are needed to solve the related algebraic conditions of unprecedented size, involving 10^17 terms (these techniques will have other applications afterwards and are of potential interest for commercial computer algebra systems as well); hence, to make a statement whether 4-dimensional integrable solitonic systems exist, and thus to contribute to a deeper understanding of the integrability of PDEs. This research will be done in collaboration with leading international experts in this area from the TU Berlin and LMU Munich.