Final Report Summary - FROM-PDE (Frobenius Manifolds and Hamiltonian Partial Differential Equations)
The theory of integrable systems is a rapidly developing branch of mathematics that brings together many research areas such as analysis, algebra, geometry. It is widely used in applied mathematics for developing analytic tools in the study of nonlinear wave processes. It proves to be helpful in various constructions of modern algebraic and differential geometry. Recent discoveries of physicists reveal deep connections of classical integrable systems with quantum field theory. These connections already proved to be important in applications to algebraic and symplectic geometry and topology. One of the starting points of the FroM-PDE project is to apply ideas from quantum field theory to the study of integrable partial differential equations. One of the goals was to develop new approaches to the long-standing problem of classification of integrable partial differential equations and, in particular, to clarify surprising connections of the classification problem with topology of Deligne - Mumford moduli spaces of algebraic curves. Another fundamental problem is in the theory of perturbations of integrable systems. What part of properties of solutions to an integrable equation survives also after a non-integrable perturbation? New trends in scientific computing based on the theory of integrable systems were also envisaged. In particular, one of the research lines of the project is in developing new methods in the extremely demanding numerical study of nonlinear dispersive partial differential equations, and to provide state of the art computational techniques for algebraic geometry.
In all these main research directions a significant progress has been achieved. The geometric theory of the so-called Frobenius manifolds created by B.Dubrovin proved to be instrumental in the construction of integrable systems of topological type (in the literature also called Dubrovin - Zhang hierarchies). New integrability tests for weakly dispersive equations were proposed by applying techniques developed in the study of the classification problem. It has been shown, by carefully organized numerical experiments and, in certain cases, by rigorous mathematical proofs that solutions to weakly dispersive Hamiltonian partial differential equations considered as perturbations of systems of hydrodynamic type at the point of phase transition from regular behaviour to rapid oscillations can be represented by just few special functions that turned out to be particular Painlevé transcendents. This justifies the Universality Conjectures formulated by Dubrovin et al. in 2006-2008. On the numerical side, modern methods of scientific computing as massive parallel algorithms were successfully applied to and optimized for nonlinear dispersive partial differential equations. It was also possible to provide the first fully numerical approach to algebraic curves.
In all these main research directions a significant progress has been achieved. The geometric theory of the so-called Frobenius manifolds created by B.Dubrovin proved to be instrumental in the construction of integrable systems of topological type (in the literature also called Dubrovin - Zhang hierarchies). New integrability tests for weakly dispersive equations were proposed by applying techniques developed in the study of the classification problem. It has been shown, by carefully organized numerical experiments and, in certain cases, by rigorous mathematical proofs that solutions to weakly dispersive Hamiltonian partial differential equations considered as perturbations of systems of hydrodynamic type at the point of phase transition from regular behaviour to rapid oscillations can be represented by just few special functions that turned out to be particular Painlevé transcendents. This justifies the Universality Conjectures formulated by Dubrovin et al. in 2006-2008. On the numerical side, modern methods of scientific computing as massive parallel algorithms were successfully applied to and optimized for nonlinear dispersive partial differential equations. It was also possible to provide the first fully numerical approach to algebraic curves.