Invariant tori for the cubic Szegö equation

By Patrick Gérard, Sandrine Grellier. 2010

63 citations recorded by the source corpus.

AbstractWe continue the study of the following Hamiltonian equation on the Hardy space of the circle, $$i\partial_tu=\Pi(|u|^2u),$$ where Π denotes the Szegö projector. This equation can be seen as a toy model for totally non dispersive evolution equations. In a previous work, we proved that this equation admits a Lax pair, and that it is completely integrable. In this paper, we construct the action-angle variables, which reduces the explicit resolution of the equation to a diagonalisation problem. As a consequence, we solve an inverse spectral problem for Hankel operators. Moreover, we establish the stability of the corresponding invariant tori. Furthermore, from the explicit formulae, we deduce the classification of orbitally stable and unstable traveling waves.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/874db8434a99b8660483bf7f25c0522c6799e5e0

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Citations in the corpus

An explicit formula for the cubic Szegö equation (2013, unknown): “In [3], action angle variables were introduced on generic subsets of the phase space, and on open dense subsets of the finite rank submanifolds.”
On the Integrability of the Benjamin‐Ono Equation on the Torus (2019, unknown): “[8], [9], [10] and references therein.”
Remarks on the Gibbs measures for nonlinear dispersive equations (2014, methodology): “— Our approach can also be applied to the Szegö equation, introduced and studied by Gérard-Grellier [25, 26, 27].”