Prerequisites: some knowledge of classical mechanics will be useful.
Contents:
Integrable Systems: 2 cycles 40 hours
The first part of the course is an introduction to the modern theory of integrable systems and its fundamental concepts.
• 1.1 Short review of the classical theory of finite-dimensional integrable systems
• 1.2 Bi-Hamiltonian Geometry and Lax pair
• 1.3 The Toda system
• 1.4 The Korteweg de Vries equation: direct and inverse scattering on the line and soliton gas
This part of the course is classical, based on my lecture notes and the following book
• Novikov, S.; Manakov, S. V.; Pitaevski, L. P.; Zakharov, V. E. Theory of solitons. The inverse scattering method. Translated from the Russian. Contemporary Soviet Mathematics. New York, 1984. xi+276 pp. ISBN: 0-306-10977-8
The second part of the course is an exploration of the generalized hydrodynamic equations. Generalized Hydrodynamics extends the usual Euler hydrodynamics to integrable systems. These equations have been derived for several integrable equations, from two different perspectives. Zakharov obtained the equations in his formulation of the kinetic theory of solitons 1971. The same equations have emerged in 2016 in the statistical mechanics of integrable systems in the thermodynamic limit. A rigorous proof of these equations exists only for the box-ball model (discrete in space and time) and the hard rod model and there is a road-map for a proof for the Toda lattice. We will explore what has been done for integrable partial differential equations.
References:
• Soliton gas: theory, numerics, and experiments P.Suret, S.Randoux, A.Gelash, D.Agafontsev, B.Doyon, G.El, Phys. Rev. E 109 (2024), no. 6, Paper No. 061001, 35 pp.
• Generalized hydrodynamics of the KdV soliton gas. T. Bonnemain, B. Doyon, G. A. El https://arxiv.org/pdf/2203.08551
• Law of Large Numbers and Central Limit Theorem for random sets of solitons of the focusing nonlinear Schr¨odinger equation. M. Girotti, T.Grava, K. D. T-R McLaughlin, J. Najnudel, Duke Math Journal 2026. Preprint https://arxiv.org/pdf/2411.17036