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Asymptotically full measure sets of almost-periodic solutions for the NLS equation

Luca Biasco, Livia Corsi, Guido Gentile, Michela Procesi

TL;DR

This work develops a PDE-appropriate KAM framework for the nonlinear Schrödinger equation on the circle with a smooth convolution potential and Gevrey data. By combining a quantitative Moser counterterm theorem with a bi-Lipschitz parameterization of initial data by linear solutions, it constructs an asymptotically full measure set of small-amplitude time almost-periodic solutions whose hulls are invariant tori, organized into a Cantor foliation of the phase space. The results establish Lyapunov statistical stability of the origin for many small data and show that, for a large (and in some cases full) measure of potentials, a large realistic set of initial data remains on invariant tori for all times. The key methodological contributions include tree-graph expansions, explicit control of small divisors via weak Bryuno conditions, and Lipschitz extensions that enable a robust infinite-dimensional KAM-type persistence result. Collectively, these findings extend KAM theory to a PDE setting with parameters, illuminating the global, measure-theoretic regularity of near-integrable infinite-dimensional Hamiltonian dynamics and providing a Cantor-structured foliation of the phase space by invariant tori.

Abstract

We study the dynamics of solutions for a family of nonlinear Schroedinger equations on the circle, with a smooth convolution potential and Gevrey regular initial data. Our main result is the construction of an asymptotically full measure set of small-amplitude time almost-periodic solutions, which are dense on invariant tori. In regions corresponding to positive actions, we prove that such maximal invariant tori are Banach manifolds, which provide a Cantor foliation of the phase space. As a consequence, we establish that, for many small initial data, the Gevrey norm of the solution remains approximately constant for all time and hence the elliptic fixed point at the origin is Lyapunov statistically stable. This is first result in KAM Theory for PDEs that regards the persistence of a large measure set of invariant tori and hence may be viewed as a strict extension to the infinite dimensional setting of the classical KAM theorem.

Asymptotically full measure sets of almost-periodic solutions for the NLS equation

TL;DR

This work develops a PDE-appropriate KAM framework for the nonlinear Schrödinger equation on the circle with a smooth convolution potential and Gevrey data. By combining a quantitative Moser counterterm theorem with a bi-Lipschitz parameterization of initial data by linear solutions, it constructs an asymptotically full measure set of small-amplitude time almost-periodic solutions whose hulls are invariant tori, organized into a Cantor foliation of the phase space. The results establish Lyapunov statistical stability of the origin for many small data and show that, for a large (and in some cases full) measure of potentials, a large realistic set of initial data remains on invariant tori for all times. The key methodological contributions include tree-graph expansions, explicit control of small divisors via weak Bryuno conditions, and Lipschitz extensions that enable a robust infinite-dimensional KAM-type persistence result. Collectively, these findings extend KAM theory to a PDE setting with parameters, illuminating the global, measure-theoretic regularity of near-integrable infinite-dimensional Hamiltonian dynamics and providing a Cantor-structured foliation of the phase space by invariant tori.

Abstract

We study the dynamics of solutions for a family of nonlinear Schroedinger equations on the circle, with a smooth convolution potential and Gevrey regular initial data. Our main result is the construction of an asymptotically full measure set of small-amplitude time almost-periodic solutions, which are dense on invariant tori. In regions corresponding to positive actions, we prove that such maximal invariant tori are Banach manifolds, which provide a Cantor foliation of the phase space. As a consequence, we establish that, for many small initial data, the Gevrey norm of the solution remains approximately constant for all time and hence the elliptic fixed point at the origin is Lyapunov statistically stable. This is first result in KAM Theory for PDEs that regards the persistence of a large measure set of invariant tori and hence may be viewed as a strict extension to the infinite dimensional setting of the classical KAM theorem.

Paper Structure

This paper contains 31 sections, 16 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Context and background
  4. Outline of the paper
  5. Methods and tools
  6. Conclusions
  7. Set up and main results
  8. Almost-periodic solutions for a large measure set of potentials
  9. Almost-periodic solutions for a full measure set of potentials
  10. Lyapunov statistical stability
  11. Structure of the invariant tori
  12. A quantitative Moser counterterm theorem
  13. The set of frequencies
  14. Almost-periodic solutions to the modified equation
  15. Properties of the almost-periodic solutions
  16. ...and 16 more sections

Key Result

Theorem 2.3

Fix $s>0$, $\alpha\in(0,1)$ and $N\geq 3$, and consider the Cauchy problem nls with potential $V\in \ell^{N,\infty}(\mathds{R})$ and initial datum $W\in \mathtt g(s,\alpha)$. There exists $\varepsilon_*=\varepsilon_*(s,\alpha,N) >0$ such that, for any $\varepsilon\in (-\varepsilon_*, \varepsilon_*)$ Moreover the measures of the sets $\mathcal{G}$ and $\mathcal{T}_V$ are both asymptotically full, i

Theorems & Definitions (56)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 2.9
  • Remark 2.10
  • ...and 46 more