Invariant tori for the cubic Szego equation

• 成果类型: Article
• 署名作者: Gerard, Patrick; Grellier, Sandrine
• 署名单位: Universite de Orleans; Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-011-0342-7
• 发表日期: 2012
• 页码: 707-754
• 关键词: schrodinger; variables

摘要

We continue the study of the following Hamiltonian equation on the Hardy space of the circle, i partial derivative(t)u = Pi(vertical bar u vertical bar(2)u), where Pi denotes the Szego 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.