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Stable manifolds for all monic supercritical NLS in one dimension

Joachim Krieger, Wilhelm Schlag

TL;DR

This work proves asymptotic stability and scattering for perturbations of monic standing waves in the one-dimensional, supercritical focusing NLS by combining spectral analysis of the linearized matrix operator with a modulation/center manifold framework. The authors develop a three-pronged approach: (1) linearization with Galilei transforms and J-invariance to set up a modulated, rest-frame dynamics; (2) a delicate spectral decomposition of the linearized operator and construction of a stable subspace via orthogonality conditions; (3) a nonlinear, time-localized iteration that builds a global solution on a stable manifold and demonstrates decay and scattering to the linear Schrödinger flow. A key contribution is a rigorous matrix scattering theory for the linearized operator, yielding sharp dispersive estimates that underpin the nonlinear analysis. The results extend previous higher-dimensional/subcritical stability work to the full one-dimensional, supercritical regime, providing a codimension-one stable manifold through symmetry arguments and precise spectral control.

Abstract

We show that all supercritical monic focusing NLS in one space dimension exhibit asymptotic stability of perturbed standing waves provided the perturbations are chosen on a small hypersuface in a suitable space.

Stable manifolds for all monic supercritical NLS in one dimension

TL;DR

This work proves asymptotic stability and scattering for perturbations of monic standing waves in the one-dimensional, supercritical focusing NLS by combining spectral analysis of the linearized matrix operator with a modulation/center manifold framework. The authors develop a three-pronged approach: (1) linearization with Galilei transforms and J-invariance to set up a modulated, rest-frame dynamics; (2) a delicate spectral decomposition of the linearized operator and construction of a stable subspace via orthogonality conditions; (3) a nonlinear, time-localized iteration that builds a global solution on a stable manifold and demonstrates decay and scattering to the linear Schrödinger flow. A key contribution is a rigorous matrix scattering theory for the linearized operator, yielding sharp dispersive estimates that underpin the nonlinear analysis. The results extend previous higher-dimensional/subcritical stability work to the full one-dimensional, supercritical regime, providing a codimension-one stable manifold through symmetry arguments and precise spectral control.

Abstract

We show that all supercritical monic focusing NLS in one space dimension exhibit asymptotic stability of perturbed standing waves provided the perturbations are chosen on a small hypersuface in a suitable space.

Paper Structure

This paper contains 9 sections, 52 theorems, 729 equations.

Table of Contents

  1. Introduction
  2. The linearization, Galilei transforms, and ${\mathcal{J}}$-invariance
  3. The linearized problem and the root spaces at zero
  4. Constructing the solution: The iteration scheme
  5. The scattering theory for Schrödinger systems
  6. Scattering solutions, the resolvent, and the distorted Fourier transform.
  7. Dispersive estimates: The unweighted case
  8. Dispersive estimates: The weighted case
  9. The spectrum of the linearized NLS

Key Result

Theorem 1.1

Fix any \sigma>2 in eq:NLS and any \alpha_0>0. Let \Sigma:=\{f\in L^2({\mathbb R})\:|\: |\!|\!| f|\!|\!| <\infty\} whereThe weight \langle x\rangle in the definition of |\!|\!|\cdot|\!|\!| can be relaxed to \langle x\rangle^{\frac{1}{2}+{\varepsilon}}, but we keep it in this form for aesthetic purpo and set Then there exist a real-linear subspace {\mathcal{S}}\subset \Sigma of co-dimension five a

Theorems & Definitions (118)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • ...and 108 more