Small amplitude weak almost periodic solutions for the 1d NLS

2023/10/01 by Luca Biasco, Jessica Elisa Massetti, Michela Procesi · 1 citation
Mathematics · Physics and Astronomy · Engineering · #Advanced Mathematical Physics Problems #Nonlinear Photonic Systems #Stability and Controllability of Differential Equations

paper · doi:10.1215/00127094-2022-0089

Abstract

All the almost periodic solutions for nonintegrable PDEs found in the literature are very regular (at least C∞) and, hence, very close to quasiperiodic ones. This fact is deeply exploited in the existing proofs. Proving the existence of almost periodic solutions with finite regularity is a main open problem in KAM theory for PDEs. Here we consider the 1-dimensional NLS with external parameters and construct almost periodic solutions which have only Sobolev regularity both in time and space. Moreover, many of our solutions are so only in a weak sense. This is the first result on existence of weak (i.e., nonclassical) solutions for nonintegrable PDEs in KAM theory.

Citations

• Newton's method and periodic solutions of nonlinear wave equations
• On invariant tori of full dimension for 1D periodic NLS
• An Abstract Birkhoff Normal Form Theorem and Exponential Type Stability of the 1d NLS

Cited by

• Invariant sets through resonant normal form for infinite dimensional holomorphic vector fields

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