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Modified scattering for nonlinear Schrödinger equations with long-range potentials

Masaki Kawamoto, Haruya Mizutani

TL;DR

This work establishes a modified scattering theory for the nonlinear Schrödinger equation with a long-range linear potential and a critical long-range nonlinearity in dimensions $1 \le n \le 3$. The authors construct a mixed asymptotic profile that combines Yafaev-like linear modifiers for the potential’s long-range part with Ozawa’s nonlinear phase correction, and prove the existence of global solutions that scatter to this profile for small localized data; a Dollard-type variant is shown to be available under stronger decay. The analysis hinges on solving a Hamilton–Jacobi phase equation to define the asymptotic profile, reformulating the problem into an integral equation, and performing a careful energy-space contraction in which the remainder is split into low- and high-velocity components with distinct decay mechanisms. The results extend modified scattering to a coupled regime where both the nonlinear term and the linear potential exhibit long-range behavior, including repulsive Coulomb-type potentials, and they introduce techniques that may apply to related long-range scattering problems in nonlinear dispersive equations.

Abstract

We study the final state problem for the nonlinear Schrödinger equation with a critical long-range nonlinearity and a long-range linear potential. Given a prescribed asymptotic profile which is different from the free evolution, we construct a unique global solution scattering to the profile. In particular, the existence of the modified wave operators is obtained for sufficiently localized small scattering data. The class of potential includes a repulsive long-range potential with a short-range perturbation, especially the positive Coulomb potential in two and three space dimensions. The asymptotic profile is constructed by combining Yafaev's type linear modifier [38] associated with the long-range part of the potential and the nonlinear modifier introduced by Ozawa [29]. Finally, we also show that one can replace Yafaev's type modifier by Dollard's type modifier under a slightly stronger decay assumption on the long-range potential. This is the first positive result on the modified scattering for the nonlinear Schrödinger equation in the case when both of the nonlinear term and the linear potential are of long-range type.

Modified scattering for nonlinear Schrödinger equations with long-range potentials

TL;DR

This work establishes a modified scattering theory for the nonlinear Schrödinger equation with a long-range linear potential and a critical long-range nonlinearity in dimensions . The authors construct a mixed asymptotic profile that combines Yafaev-like linear modifiers for the potential’s long-range part with Ozawa’s nonlinear phase correction, and prove the existence of global solutions that scatter to this profile for small localized data; a Dollard-type variant is shown to be available under stronger decay. The analysis hinges on solving a Hamilton–Jacobi phase equation to define the asymptotic profile, reformulating the problem into an integral equation, and performing a careful energy-space contraction in which the remainder is split into low- and high-velocity components with distinct decay mechanisms. The results extend modified scattering to a coupled regime where both the nonlinear term and the linear potential exhibit long-range behavior, including repulsive Coulomb-type potentials, and they introduce techniques that may apply to related long-range scattering problems in nonlinear dispersive equations.

Abstract

We study the final state problem for the nonlinear Schrödinger equation with a critical long-range nonlinearity and a long-range linear potential. Given a prescribed asymptotic profile which is different from the free evolution, we construct a unique global solution scattering to the profile. In particular, the existence of the modified wave operators is obtained for sufficiently localized small scattering data. The class of potential includes a repulsive long-range potential with a short-range perturbation, especially the positive Coulomb potential in two and three space dimensions. The asymptotic profile is constructed by combining Yafaev's type linear modifier [38] associated with the long-range part of the potential and the nonlinear modifier introduced by Ozawa [29]. Finally, we also show that one can replace Yafaev's type modifier by Dollard's type modifier under a slightly stronger decay assumption on the long-range potential. This is the first positive result on the modified scattering for the nonlinear Schrödinger equation in the case when both of the nonlinear term and the linear potential are of long-range type.
Paper Structure (16 sections, 15 theorems, 170 equations)

This paper contains 16 sections, 15 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Background on the nonlinear scattering
  4. Idea of the proof
  5. Organization of the paper
  6. Preliminaries
  7. Notation
  8. Dollard decomposition
  9. Integral equation
  10. Nonlinear estimates
  11. The phase function $\Psi$
  12. Energy norm of ${\mathcal{E}}(t)$.
  13. Proof of Theorem \ref{['theorem_1']}
  14. Proof of Theorem \ref{['theorem_3']}
  15. Global existence for the Cauchy problem
  16. ...and 1 more sections

Key Result

Theorem 1.4

Let $1\le n\le 3$, $\gamma$ be as in Assumption assumption_B and Let $V$ and $u_+$ satisfy Assumptions assumption_A and assumption_B, respectively, and ${\|\widehat{u_+}\|}_{L^\infty}$ be small enough. Then there exists a unique solution $u\in C({\mathbb{R}};L^2({\mathbb{R}}^n))$ to NLS1 satisfying, for any admissible pair $(q,r)$,

Theorems & Definitions (36)

  • Remark 1.1
  • Example 1.2
  • Remark 1.3
  • Theorem 1.4: Modified scattering
  • Remark 1.5
  • Corollary 1.6: Modified wave operator
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9: Dollard type modification
  • Lemma 2.1
  • ...and 26 more