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Threshold for the existence of scattering states for nonlinear Schrödinger equations without gauge invariance

Hayato Miyazaki, Motohiro Sobajima

TL;DR

This work identifies a sharp threshold for scattering in nonlinear Schrödinger equations lacking gauge invariance, showing that in a weighted space $\mathcal{F}H^{\alpha}$ with $\alpha\ge s_c$, no nontrivial scattering states exist when the nonlinearity power satisfies $1<p<p_{\rm st}(d)$. The authors develop a test-function, weak-solution framework and a lower-bound argument tied to the Strauss exponent, demonstrating that even data with compact support can be non-scattering. The results highlight the Strauss exponent as a boundary between scattering and non-scattering behavior in the non-gauge-invariant setting, contrasting with the gauge-invariant case where scattering can persist under certain regimes. The methodology and weighted-space perspective offer a natural approach to the long-time behavior of solutions for subcritical nonlinearities and emphasize the role of structural assumptions on the nonlinearity.

Abstract

This paper is concerned with a threshold phenomenon for the existence of scattering states for nonlinear Schrödinger equations. The nonlinearity includes a non-oscillatory term of the order lower than the Strauss exponent. We show that no scattering states exist for the equation in a weighted Sobolev space. It is emphasized that our method admits initial data with good properties, such as compactly supported smooth functions. The result indicates that the Strauss exponent acts as a threshold for the power of the nonlinearity that determines whether solutions scatter or not in the weighted space.

Threshold for the existence of scattering states for nonlinear Schrödinger equations without gauge invariance

TL;DR

This work identifies a sharp threshold for scattering in nonlinear Schrödinger equations lacking gauge invariance, showing that in a weighted space with , no nontrivial scattering states exist when the nonlinearity power satisfies . The authors develop a test-function, weak-solution framework and a lower-bound argument tied to the Strauss exponent, demonstrating that even data with compact support can be non-scattering. The results highlight the Strauss exponent as a boundary between scattering and non-scattering behavior in the non-gauge-invariant setting, contrasting with the gauge-invariant case where scattering can persist under certain regimes. The methodology and weighted-space perspective offer a natural approach to the long-time behavior of solutions for subcritical nonlinearities and emphasize the role of structural assumptions on the nonlinearity.

Abstract

This paper is concerned with a threshold phenomenon for the existence of scattering states for nonlinear Schrödinger equations. The nonlinearity includes a non-oscillatory term of the order lower than the Strauss exponent. We show that no scattering states exist for the equation in a weighted Sobolev space. It is emphasized that our method admits initial data with good properties, such as compactly supported smooth functions. The result indicates that the Strauss exponent acts as a threshold for the power of the nonlinearity that determines whether solutions scatter or not in the weighted space.

Paper Structure

This paper contains 5 sections, 5 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of the main theorem
  4. Preliminary
  5. Proof of Theorem \ref{['thm:1']}

Key Result

Theorem 1.2

Let $d \geqslant 1$. Assume A1 and f:con with $1 < p < p_{\mathrm{st}}(d)$. Take $\alpha>0$ satisfying Given $u_{0} \in \mathcal{F}H^{\alpha}$, let $u \in C([0, \infty) ; L^{2})$ be a solution to nls with $u(0) = u_{0}$. The following statements hold:

Theorems & Definitions (18)

  • Definition 1.1: Solution
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 2.1
  • proof
  • ...and 8 more