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Instability of the solitary waves for the generalized derivative nonlinear Schrödinger equation in the degenerate case

Changxing Miao, Xingdong Tang, Guixiang Xu

Abstract

In this paper, we develop the modulation analysis, the perturbation argument and the Virial identity similar as those in \cite{MartelM:Instab:gKdV} to show the orbital instability of the solitary waves $\Q\sts{x-ct}\e^{ıωt}$ of the generalized derivative nonlinear Schrödinger equation (gDNLS) in the degenerate case $c=2z_0\sqrtω$, where $z_0=z_0\stsσ $ is the unique zero point of $F\sts{z;~σ}$ in $\sts{-1, ~ 1}$. The new ingredients in the proof are the refined modulation decomposition of the solution near $\Q$ according to the spectrum property of the linearized operator $\Scal_{ω, c}"\sts{\Q}$ and the refined construction of the Virial identity in the degenerate case. Our argument is qualitative, and we improve the result in \cite{Fukaya2017}.

Instability of the solitary waves for the generalized derivative nonlinear Schrödinger equation in the degenerate case

Abstract

In this paper, we develop the modulation analysis, the perturbation argument and the Virial identity similar as those in \cite{MartelM:Instab:gKdV} to show the orbital instability of the solitary waves of the generalized derivative nonlinear Schrödinger equation (gDNLS) in the degenerate case , where is the unique zero point of in . The new ingredients in the proof are the refined modulation decomposition of the solution near according to the spectrum property of the linearized operator and the refined construction of the Virial identity in the degenerate case. Our argument is qualitative, and we improve the result in \cite{Fukaya2017}.

Paper Structure

This paper contains 8 sections, 13 theorems, 152 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic properties of ${\mathcal{S}}_{\omega,c}$ and $d\left(\omega,c\right)$
  4. Geometric decomposition of $u$ and landscape of ${\mathcal{S}}$ near $Q$
  5. Properties of the linearized operator $\mathcal{L}$
  6. The equation on $\varepsilon$ variable and the dynamical estimates of the parameters
  7. Proof of Theorem \ref{['thm']}
  8. Proof of Lemma \ref{['lem:S3d']}

Key Result

Theorem 1.2

Let $\sigma\in \left(1, 2\right)$, and $c=2 z_0 \sqrt{\omega}$ where $z_0=z_0(\sigma)$ be the unique zero point in $\left(-1,1\right)$ of $F(z; \sigma)$ in eq:sig-z, then the solitary wave ${Q}_{\omega,c}\left(x-ct\right)\text{e}^{\,\text{i}\,\omega t}$ of gdnls is orbitally unstable. More precisely where $0<\lambda<\lambda_0$ and $\widetilde{\rho}\left(\lambda\right)$ is chosen such that ${\mathc

Figures (2)

  • Figure 1: The zero point $z_0\left(\sigma\right)$ of $F(z; \sigma)$ in $\left(-1,~1\right)$ for $\sigma \in \left(1,~2\right)$, and $z_0\left(\sigma\right)$ is a decreasing function as $\sigma$ increases in $(1, 2)$.
  • Figure 2: Decompostion of the $\delta$-tube ${\mathcal{U}}\left({Q}_{\omega,c}, ~\delta\right)$ up to small radiation $\varepsilon$.

Theorems & Definitions (30)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 20 more