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Scattering theory for the Gross-Pitaevskii equation in three dimensions

S. Gustafson, K. Nakanishi, T. -P. Tsai

TL;DR

This work establishes global-in-time behavior for small energy perturbations of the three-dimensional Gross–Pitaevskii equation with nonzero boundary data, showing dispersion at large times governed by the linearized flow $e^{-itH}$ where $H=\sqrt{-\Delta(2-\Delta)}$. A central technical achievement is the combination of a quadratic normal-form transform $M(u)$ with a refined bilinear multiplier framework that tolerates singular, degenerate resonances near zero frequency; this underpins both final-state scattering and initial-data scattering results. The authors prove asymptotic stability of all plane waves for the defocusing NLS under such disturbances and construct the inverse-scattering map in the renormalized energy space $F_1$, leveraging a careful geometric analysis of resonant regions and a novel singular bilinear estimate (Lemma sbil). The results provide a rigid, quantitative framework for nonlinear scattering in GP with nonzero boundary data and advance the understanding of how energy and renormalized structures govern long-time dynamics in dispersive systems.

Abstract

We study global behavior of small solutions of the Gross-Pitaevskii equation in three dimensions. We prove that disturbances from the constant equilibrium with small, localized energy, disperse for large time, according to the linearized equation. Translated to the defocusing nonlinear Schrödinger equation, this implies asymptotic stability of all plane wave solutions for such disturbances. We also prove that every linearized solution with finite energy has a nonlinear solution which is asymptotic to it. The key ingredients are: (1) some quadratic transforms of the solutions, which effectively linearize the nonlinear energy space, (2) a bilinear Fourier multiplier estimate, which allows irregular denominators due to a degenerate non-resonance property of the quadratic interactions, and (3) geometric investigation of the degeneracy in the Fourier space to minimize its influence.

Scattering theory for the Gross-Pitaevskii equation in three dimensions

TL;DR

This work establishes global-in-time behavior for small energy perturbations of the three-dimensional Gross–Pitaevskii equation with nonzero boundary data, showing dispersion at large times governed by the linearized flow where . A central technical achievement is the combination of a quadratic normal-form transform with a refined bilinear multiplier framework that tolerates singular, degenerate resonances near zero frequency; this underpins both final-state scattering and initial-data scattering results. The authors prove asymptotic stability of all plane waves for the defocusing NLS under such disturbances and construct the inverse-scattering map in the renormalized energy space , leveraging a careful geometric analysis of resonant regions and a novel singular bilinear estimate (Lemma sbil). The results provide a rigid, quantitative framework for nonlinear scattering in GP with nonzero boundary data and advance the understanding of how energy and renormalized structures govern long-time dynamics in dispersive systems.

Abstract

We study global behavior of small solutions of the Gross-Pitaevskii equation in three dimensions. We prove that disturbances from the constant equilibrium with small, localized energy, disperse for large time, according to the linearized equation. Translated to the defocusing nonlinear Schrödinger equation, this implies asymptotic stability of all plane wave solutions for such disturbances. We also prove that every linearized solution with finite energy has a nonlinear solution which is asymptotic to it. The key ingredients are: (1) some quadratic transforms of the solutions, which effectively linearize the nonlinear energy space, (2) a bilinear Fourier multiplier estimate, which allows irregular denominators due to a degenerate non-resonance property of the quadratic interactions, and (3) geometric investigation of the degeneracy in the Fourier space to minimize its influence.

Paper Structure

This paper contains 31 sections, 7 theorems, 292 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Normal forms
  4. Final data problem
  5. Initial data problem
  6. Decay property by the weighted norm
  7. Estimates on the normal form
  8. Other normal forms
  9. Initial condition
  10. Estimates without weight
  11. Estimates without phase derivative
  12. Estimates with phase derivative in bilinear terms
  13. Singular bilinear multiplier with Strichartz
  14. Spatial integration of phase
  15. Time integration of phase
  16. ...and 16 more sections

Key Result

Theorem 1.1

There exists $\delta>0$ such that: For any $u(0)\in H^1(\mathbb{R}^3)$ satisfying we have a unique global solution $\psi=1+u$ of GP such that $v := \mathop{\mathrm{Re}} u + i U\mathop{\mathrm{Im}} u$ satisfies $e^{itH}v\in C(\mathbb{R};{\langle x \rangle}^{-1}H^1(\mathbb{R}^3))$ and for some $v_+\in{\langle x \rangle}^{-1}H^1(\mathbb{R}^3)$, as $t\to\infty$. Moreover, we have $E_1(\psi)=\|{\lang

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 3.1
  • proof : Proof of Theorem \ref{['thm:map']}
  • ...and 6 more