Further progress in integrable theory, as well as possible applications to real physical problems, will strongly depend on understanding the asymptotic integrability of classical and quantum systems depending on a parameter; that is to say, integrable behaviour is only realized in the regimes where the parameter tends to certain limiting values. The Whitham averaging technique proved to be an efficient tool not only in the asymptotic theory of nonlinear waves but also in quantum field theory and in the theory of random matrices. Geometric and analytic foundations of the averaging methods rely upon the study of behaviour of trajectories of integrable systems in the neighborhood of a finite dimensional invariant manifold. The results of this study will also be used in applying dispersive integrable limits to weak turbulence.
In the studies of the borderline between integrable and non-integrable behaviour in evolutionary PDEs, it has long been suspected that semiclassical problems for non-integrable equations are somehow reduced to integrable problems. Further numerical evidences supporting this conjecture were found in the recent paper by J.Bergamin, S.Kamvissis and T.Bountis. Among our plans is the clarification (with rigorous statement and proofs) of this issue. We will also study numerically the properties of the PDEs obtained by a truncation of the perturbative expansions of the integrable systems of infinite order.
Preliminary numerical experiments developed by P.Lorenzoni suggest that in such approximately integrable systems an integrable behaviour can be realized within a certain range of the initial data. This analysis will be applied to the problem of dispersive regularization of hyperbolic systems of conservation laws. Certain geometric and analytic aspects of fluid dynamics will constitute an important part of our research project. In particular measure-valued solutions for a family of evolutionary PDEs in one, two and three spatial dimensions will be studied. The invariant manifold of measured-valued solutions under consideration arises from an interesting family of momentum maps for the action of diffeomorphisms on lower dimensional subspaces of three-dimensional space. Also the powerful methods, developed by Kamvissis based on the Riemann-Hilbert problems and their deformations will be applied to producing asymptotics for solutions of integrable evolution equations and also problems arising from random matrices.
Dyson has subjected a random Hermitian matrix in the Gaussian Unitary Ensemble to an Ornstein-Uhlenbeck process. In the large size limit, what is the random motion of the eigenvalues of the matrices. This has led Prähofer-Spohn to the so-called Airy process for the top-largest eigenvalue. It is a new stationary random process, with continuous sample paths, with stationary distribution given by the Tracy-Widom distribution (Johansson). Its joint distribution at different times is related to a Fredholm determinant of an extended Airy kernel (Tracy-Widom) or to Hermitian matrix models in a chain (Adler-van Moerbeke). Very little is known about this process and the processes corresponding to the other eigenvalues; also nothing is known about many other Dyson-type motions. Various random growth models, properly rescaled, have Airy Process limiting distributions, as shown by Johansson. We also plan to get a deeper grasp of the statistical properties of one-dimensional growth processes and, in parallel, of certain tiling problems in the plane.