Wave Turbulence and Complete Integrability

On a Damped Szegö Equation (With an Appendix in Collaboration With Christian Klein)

“…Let us mention that some introductory material to the Szegő equation can be found in [9], and in [10], where the results of this paper were announced.…”

“…This question, raised by Bourgain in [1], [2] for the defocusing nonlinear Schrödinger equation on the torus, led to several contributions constructing solutions with a small initial Sobolev norm of high regularity and a big Sobolev norm at some later time, see [4], [5], [8], [13], [16], [19], [15], [14], [12]. The actual existence of unbounded trajectories was proved in [21], [17], [18], [7], [23], [24], [3], [22], [9]. In this paper, we intend to study a case where a weak damping can promote unbounded trajectories in Sobolev spaces with high regularity.…”

“…An important property of equation 1.1is the existence of a Lax pair structure, leading to action-angle variables [6]. On the other hand, the conservation laws do not control high Sobolev regularity of the solutions, allowing for long term infinitely many transitions between low and high frequencies, as proved in [7] -see also the lecture notes [9]. However, such solutions are very unstable.…”

Steady states in dual-cascade wave turbulence

“…In spite of a very broad range of turbulent systems for which differential approximation closures were successfully used, there exist examples for which such models are not yet known. These include the cases of the so-called integrable wave turbulence [17,18].…”