test_moissonnage_selune
type: Agent

Study of a nonlinear, non-dispersive, completely integrable equation and its perturbations
type: Dataset
contact point: Organization
  fn: CCSD
keyword: action-angle-coordinates
keyword: lax-pair
keyword: szego-equation
keyword: non-linear-wave-equation
keyword: equation-de-szego
keyword: soliton
keyword: renormalisation-group-method
keyword: operateur-de-hankel
keyword: coordonnees-action-angle
keyword: paire-de-lax
keyword: theses
keyword: infoeu-reposemanticsdoctoralthesis
keyword: soliton-resolution
keyword: hankel-operators
keyword: mathmath-gmmathematics-mathgeneral-mathematics-mathgm
keyword: methode-du-groupe-de-renormalisation
keyword: equation-des-ondes-non-lineaire
landing page: https://theses.hal.science/tel-00834518
description: In this Ph.D. thesis, we study the Szegö equation on the real lineas well as its perturbations.It was recently introduced by Gérard and Grellier as a toy model of a non-lineartotally non dispersive equation. The Szegö equation appears naturally in the study of thenon-linear Schrödinger equation (NLS) in super-critical situations where dispersion lacks,for example, when one considers NLS on the Heisenberg group. Consequently, one of themotivations of this Ph.D. thesis is fi nding new results for the Szegö equation in hope thatthey could be eventually used in the context of the non-linear Schrödinger equation.Our first result is a classification of the solitons of the Szegö equation. We show thatthey are all rational functions with one simple pole. In addition, we prove the orbitalstability of solitons.The Szegö equation has the remarkable property of being completely integrable. Thisallows us to find an explicit formula for solutions. We obtain three applications of thisformula. (A) We prove soliton resolution for solutions which are generic rational functions.(B) We construct an example of non-generic solution whose high Sobolev norms grow toinfinity over time. (C) We find generalized action-angle variables when restricting the Szegöequation to a finite dimensional sub-manifold. In particular, this yields that most of thetrajectories of the Szegö equation are spirals around toroidal cylinders.Since the Szegö equation is completely integrable, it is natural to study its perturbationsand deduce new properties of such perturbations from the known results for the Szegöequation. One perturbation of the Szegö equation is a non-linear wave equation(NLW) with small initial data.We prove that the Szegö equation is the first order approximation of NLW. More precisely,if an initial condition of NLW is small and supported only on non-negative frequencies, thenthe corresponding solution can be approximated by the solution of the Szegö equation, fora long time. We then construct …
identifier: NNT: 2011PA112182
issued: 2026-05-10T17:16:36.377155
language: en
modified: 2026-05-10T17:16:36.377161

Study of a nonlinear, non-dispersive, completely integrable equation and its perturbations
type: Distribution
access URL: https://theses.hal.science/tel-00834518
format: HTML
issued: 2026-05-10T17:16:36.403849
modified: 2026-05-10T17:16:36.350183