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Mathematics > Analysis of PDEs

Title: 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 $\Q\sts{x-ct}\e^{\i\omega t}$ of the generalized derivative nonlinear Schr\"odinger equation (gDNLS) in the degenerate case $c=2z_0\sqrt{\omega}$, where $z_0=z_0\sts{\sigma} $ is the unique zero point of $F\sts{z;~\sigma}$ 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 $\Scal_{\omega, c}"\sts{\Q}$ and the refined construction of the Virial identity in the degenerate case. Our argument is qualitative, and we improve the result in \cite{Fukaya2017}.
Comments: 33 pages, 2 figures, all comments are welcome
Subjects: Analysis of PDEs (math.AP)
Cite as: arXiv:1803.06451 [math.AP]
  (or arXiv:1803.06451v1 [math.AP] for this version)

Submission history

From: Guixiang Xu [view email]
[v1] Sat, 17 Mar 2018 03:52:15 GMT (52kb,D)