Computer algebra and the integrability of continuous and discrete systems

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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.