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Construction of a multi-soliton-like solutions for non-integrable Schrödinger equations with non-trivial far field

Jordan Berthoumieu

Abstract

This article provides a naturel sequel of previous works [6, 4] regarding the stability of travelling waves for a general one-dimensional Schrödinger equation (N LS) with non-zero condition at infinity. The aim of this article is twofold. First, we prove the asymptotic stability of well-prepared chains of dark solitons and secondly, we construct an asymptotic N -soliton-like solution, which is an exact solution of (N LS), the large-time dynamics of which is similar to a decoupled chain of solitons.

Construction of a multi-soliton-like solutions for non-integrable Schrödinger equations with non-trivial far field

Abstract

This article provides a naturel sequel of previous works [6, 4] regarding the stability of travelling waves for a general one-dimensional Schrödinger equation (N LS) with non-zero condition at infinity. The aim of this article is twofold. First, we prove the asymptotic stability of well-prepared chains of dark solitons and secondly, we construct an asymptotic N -soliton-like solution, which is an exact solution of (N LS), the large-time dynamics of which is similar to a decoupled chain of solitons.

Paper Structure

This paper contains 15 sections, 20 theorems, 182 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Dynamical setting
  3. Travelling wave solutions and chain of well-prepared solitons
  4. Main results
  5. Hydrodynamical setting
  6. Existence of a solution in different frameworks
  7. Estimate in a neighborhood of a chain
  8. Asymptotic stability of a well-prepared chain
  9. From orbital to asymptotic stability
  10. Rigidity property between the travelling waves
  11. Proof of the algebraic decay
  12. Asymptotic stability in the original framework
  13. Construction of an asymptotic $N$-soliton
  14. Reduction of the problem
  15. Proof of the uniform estimate

Key Result

Theorem 1.1

Let $u_0\in \mathcal{X}(\mathbb{R})$. Take $f$ in $C^2(\mathbb{R})$ satisfying hypothèse de croissance sur F minorant intermediaire. In addition, assume that there exist $\alpha_1\geq 1$ and $C_0 >0$ such that for all $\rho\geq 1$, If $\alpha_1 > \frac{3}{2}$, assume moreover that there exists $\alpha_2\in [ \alpha_1-\frac{1}{2},\alpha_1]$ such that for $\rho\geq 2$, $C_0 \rho^{\alpha_2} \leq F(\

Figures (1)

  • Figure 1: The modulus $\sqrt{1-\eta_{\mathfrak{c},\mathfrak{a}}}$ of a well-prepared chain of three solitons, with speeds $c_1 < c_2 < c_3$, and $(a_1,a_2,a_3)\in\mathrm{Pos}_3(L)$.

Theorems & Definitions (48)

  • Theorem 1.1: Gallo1
  • Theorem 1.2: Bert2Bert3
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 2.1: Gallo3
  • ...and 38 more