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High-order long-time asymptotics for small solutions to the one-dimensional nonlinear Schrödinger equation

Jacek Jendrej, Tony Salvi

Abstract

We investigate the global well-posedness and modified scattering for the one-dimensional Schrödinger equation with gauge-invariant polynomial nonlinearity. For small localized initial data of finite energy in a low-regularity class, we establish global existence of solution together with persistence of the localization of the associated profile. We further provide a rigorous derivation of the asymptotic expansion at arbitrary order of such solutions, taking into account long-range effects induced by the cubic component of the nonlinearity. Our analysis relies on the space-time resonance method.

High-order long-time asymptotics for small solutions to the one-dimensional nonlinear Schrödinger equation

Abstract

We investigate the global well-posedness and modified scattering for the one-dimensional Schrödinger equation with gauge-invariant polynomial nonlinearity. For small localized initial data of finite energy in a low-regularity class, we establish global existence of solution together with persistence of the localization of the associated profile. We further provide a rigorous derivation of the asymptotic expansion at arbitrary order of such solutions, taking into account long-range effects induced by the cubic component of the nonlinearity. Our analysis relies on the space-time resonance method.
Paper Structure (14 sections, 28 theorems, 215 equations)

This paper contains 14 sections, 28 theorems, 215 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement and funding
  3. Notation and common results
  4. Main ideas of the proof
  5. Preliminary computations
  6. Formulation of the equations
  7. Basic error estimates
  8. Local existence result
  9. Global existence and bootstrap
  10. Asymptotic analysis
  11. Asymptotics for the modified profile
  12. Asymptotics for the profile
  13. Asymptotics for the solution
  14. Explicit form of the expansion at order 2

Key Result

Theorem 1.1

There exists $\varepsilon_0>0$ such that, if $||u_1||_{H^{1,0}_x}+||f_1||_{H^{0,2N+1}_x}<\varepsilon_0$, then there exists a unique global solution to eq:NLS, such that $u\in C([1,\infty),H^1_x)$. Moreover this solution has the sharp decay and admits an asymptotic expansion of order $N$ under the form where for $0\leq p\leq N$, $0\leq k\leq 2p$, $u_{p,k}\in W^{2(N-p),\infty}_x$, $\varphi\in W^{

Theorems & Definitions (62)

  • Theorem 1.1
  • Lemma 3.1
  • Remark 5.3
  • Remark 5.5
  • Lemma 5.6
  • proof : Proof of Lemma \ref{['lem:phasenoxi']}
  • Lemma 5.9
  • proof
  • Remark 5.10
  • Remark 5.11
  • ...and 52 more