KAM tori for perturbations of the defocusing NLS equation
By: Massimiliano Berti, Thomas Kappeler & Riccardo Montalto
Contributors: Kappeler, Thomas, 1953- author; Montalto, Riccardo, author
Series: Astérisque -- 403; Astérisque -- 403.
Publisher: Société mathématique de France
Published: 2018
Place of publication: Paris
Language: eng
Number of pages: 148
ISBN-13: 9782856298923
Subjects: Perturbation (Mathematics); Kolmogorov-Arnold-Moser theory; Schrödinger equation; Implicit functions; 31.45 partial differential equations
Other titles: Large KAM tori for perturbations of the defocusing NLS equation
Description: We prove that small, semi-linear Hamiltonian perturbations of the defocusing nonlinear Schrödinger (dNLS) equation on the circle have an abundance of invariant tori of any size and (finite) dimension which support quasi-periodic solutions. When compared with previous results the novelty consists in considering perturbations which do not satisfy any symmetry condition (they may depend on x in an arbitrary way) and need not be analytic. The main difficulty is posed by pairs of almost resonant dNLS frequencies. The proof is based on the integrability of the dNLS equation, in particular the fact that the nonlinear part of the Birkhoff coordinates is one smoothing. We implement a Newton-Nash-Moser iteration scheme to construct the invariant tori. The key point is the reduction of linearized operators, coming up in the iteration scheme, to 2{u00D7}2 block diagonal ones with constant coefficients together with sharp asymptotic estimates of their eigenvalues.
Notes: Correct title per erratum: Large KAM tori for perturbations of the defocusing NLS equation.

Includes bibliographical references (pages 145-148).

Abstract also in French.