Computer algebra and the integrability of continuous and discrete systems

在数学和物理学中，孤子是由介质中的非线性效应和色散效应之间的微妙平衡引起的一种自增强孤波。物理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。孤子在工业上与通过光纤系统的传输有关，光纤系统提高了光通信的性能。具有孤子解的偏微分方程(PDE)表现出许多特征性质，如存在由简单孤子解转化为更复杂孤子解的所谓贝克伦德变换的存在，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此类偏微分方程组的所有这些性质：由满足所谓面关系的n维离散面建立的离散(n，1)维网格的一致性。具有孤子解的偏微分方程组及其Baecklund变换和叠加原理可以通过适当的极限程序从这些(n，1)维网格中得到。这个提议的一个目的是研究这种面关系(例如，确定主要类别的三维和四维面关系)，并为此开发计算机代数算法和改进技术，以解决涉及10^17项的空前大的相关代数条件(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，陈述是否存在4维可积孤子系统，从而有助于更深入地理解偏微分方程组的可积性。这项研究将与柏林工业大学和LMU慕尼黑大学在这一领域的领先国际专家合作进行。

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.