Patrick Gérard

A review of the cubic Szegö equation

The cubic Szegö equation is an integrable Hamiltonian system which can be obtained from some one dimensional simple model of nonlinear wave interaction via time averaging. The specificity of its integrable structure is that it allows generic long time transitions to high frequencies. I will review the main properties of this equation, starting with its Lax pair structure, which connects its dynamics to spectral theory of Hankel matrices, and giving an idea of its action angle variables and of their singularities, which are the key to the construction of long time oscillatory trajectories. This introductory talk is based on a series of joint works
with Sandrine Grellier (Orleans).