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Orbital Stability of Soliton for the Derivative Nonlinear Schrödinger Equation in the $L^2$ Space

Yiling Yang, Engui Fan, Yue Liu

Abstract

In this paper, we establish the orbital stability of the 1-soliton solution for the derivative nonlinear Schrödinger equation under perturbations in $L^2(\mathbb{R})$. We demonstrate this stability by utilizing the Bäcklund transformation associated with the Lax pair and by applying the first conservation quantity in $L^2(\mathbb{R}).$

Orbital Stability of Soliton for the Derivative Nonlinear Schrödinger Equation in the $L^2$ Space

Abstract

In this paper, we establish the orbital stability of the 1-soliton solution for the derivative nonlinear Schrödinger equation under perturbations in . We demonstrate this stability by utilizing the Bäcklund transformation associated with the Lax pair and by applying the first conservation quantity in
Paper Structure (5 sections, 11 theorems, 165 equations, 1 figure)

This paper contains 5 sections, 11 theorems, 165 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. From a perturbed 1-soliton solution to a small solution at $t = 0$
  4. From a small solution back to perturbed 1-soliton solution
  5. Proof of stability theorem

Key Result

Proposition 1.1

KillipDNLS The DNLS equation DNLS is globally well-posed in $H^s(\mathbb{R})$, for every $s\geq 0$. More precisely, given any $T>0$, an initial data $q(0)\in L^2(\mathbb{R})$ and a sequence of Schwartz-class initial data $q_n(0)$ converging to $q(0)$ in $L^2(\mathbb{R})$, then the sequence of corres

Figures (1)

  • Figure 1: The Bäcklund transformation approach for the stability of the 1-soliton within a small $L^2$-neighborhood

Theorems & Definitions (22)

  • Proposition 1.1
  • Remark 1.1
  • Theorem 1.1
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Corollary 3.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 12 more