Final Report Summary - HYDRON (Near Hydrodynamic Type Systems in 2 +1 Dimensions) Non-linear partial differential equations (PDEs) typically possess solutions that blow up unexpectedly. If such an equation describes the evolution of a physical system, the exploration of such critical events is of uttermost importance. Well known examples include shock waves, freak waves in the ocean, as well as space-time singularities in general relativity. There is a special class of PDEs for which blow up of solutions is a generic feature, the so called first order quasilinear equations, also known as systems of hydrodynamic type. Such systems appear as limiting cases of other PDEs, which often yield a more realistic description of the physical system in question. This motivates the study of systems that are deformations of hydrodynamic-type ones, essentially looking for a way to reverse the limiting procedure. Among such deformations are examples from the class of completely integrable PDEs, which can be studied with special analytic methods and include the prominent, physically important soliton equations. The aim of this project was to study hydrodynamic-type systems and corresponding deformations in 2+1 dimensions. One way to explore the relationship between hydrodynamic-type systems and soliton equations is via bidifferential calculus. Integrable PDEs, as well as integrable partial difference equations, can be thought of as different realisations of equations in this framework. So far this does not extend to a treatment of hydrodynamic-type systems and only little progress has been made in this respect during this project. But it turned out that, in several (2+1)-dimensional integrable systems, hydrodynamic-type systems can appear in a different way than via the aforementioned limiting procedure. In the bidifferential calculus framework, equations that formally resemble a (matrix) Riemann equation have been considered. Realizations of these equations include a semi-discrete and a fully discrete regularisation of the (matrix) Riemann equation as the simplest cases. Solutions of a 'Riemann system' are also solutions of an associated soliton equation that arises as a consistency condition. For this equation a binary Darboux transformation method generates exact solutions. In the particular case of a three-dimensional generalisation of the non-linear Schrödinger equation (2+1 NLS), the hydrodynamic-type limit turned out to be not integrable by the known methods for hydrodynamic-type systems. However, being a limit of an integrable equation, such a system is expected to be integrable in some sense. Further exploration led to the discovery of a new family of three-dimensional quasilinear systems. It has been made precise in what sense this family is indeed integrable. Besides this, it has also been shown that the 'Riemann system' associated with 2+1 NLS is the origin of its breaking and blow-up soliton-type solutions. Moreover, within this project a special hydrodynamic-type system – one of the so called Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations – has been explored. In physics these equations arise as associativity conditions of certain algebras related to topological field theories. Although from a mathematical point of view they form a quasilinear system, hence a hydrodynamic-type system, they have many characteristics of a soliton equation. For instance, their solutions do not blow up in finite time. This has found an explanation now, since it has been possible to construct a system equivalent to the WDVV equation, which lies in the framework of deformations of hydrodynamic-type systems.
