Integrable systems, Frobenius manifolds and nonlinear waves

Research Topics


Integrable evolutionary differential equations were discovered at the end of the 60's in the study of dispersive waves. It has been realized afterwards that integrable systems of analytical mechanics, and also classical Painleve' equations can be embedded into the general theory of integrable systems as particular reductions of integrable PDEs, and their study received a brand new impulse by the use of methods introduced in the analysis of PDEs(e.g., the Lax method).
Mathematical progresses of the theory of integrable systems was related to Riemann - Hilbert boundary value problems, to algebraic and differential geometry, to infinite dimensional Lie algebras, and to quantum groups. As an example, we recall the striking solution of the classical Riemann - Schottky problem via the proof (due to Dubrovin, Arbarello - De Concini and Shiota) of Novikov's conjecture about the solutions of soliton equations.
An important general approach to integrable systems was introduced by F. Magri. It is based on the geometry of bihamiltonian systems. After the discovery in the early 90s of the fundamental role of the Korteweg-de Vries equation (KdV) in the matrix and topological models of two-dimensional gravity, and, in particular, after the solution, by E. Witten and M. Kontsevich, of the old problem of the topology of the moduli spaces of algebraic curves in terms of KdV hierarchy, the theory of integrable systems has become one of the unifying principles in the mathematics of the beginning of the 21st century. Further developments of two-dimensional topological field theory supplied the theory of integrable systems with a new tool based on the geometric and analytic theory of Frobenius manifolds. They were introduced by B. Dubrovin as the geometrical setting for the so-called WDVV equations of associativity, but, subsequently, have become a common paradigma for rather distant branches of mathematics, such as Gromov - Witten invariants of symplectic manifolds, singularity theory, isomonodromy deformations, and the theory of reflection groups and their extensions.
Actually, various fundamental problems of these branches of mathematics, such as the problem of Gromov - Witten invariants at higher genera and the problem of integrability in four dimensional supersymmetric field theories, require a new approach to the problem of classification of integrable systems. Our approach to this problem is naturally based on the theory of Frobenius manifolds. This theory suggests to classify integrable bihamiltonian hierarchies of evolutionary PDEs which possess a tau-function and Virasoro invariance. The KdV hierarchy and the Drinfeld-Sokolov hierarchies corresponding to simply laced simple Lie algebras are examples of these hierarchies, but this list can be strongly extended. A closely related topic is the study, by means of differential geometrical and algebro geometrical techniques, of finite-dimensional integrable Hamiltonian systems (especially those arising as suitable reductions of infinite-dimensional ones). Among the many open problems of this theory we can mention the properties of the action-angle variables in the framework of special Kaehler geometry and the theory of Frobenius manifolds. The solution of these problems is also important for applications of integrable systems to the Quantum Theory.


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(partially funded by MISGAM and ENIGMA)