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arXiv · 1803.06451

Instability of the solitary waves for the generalized derivative nonlinear Schr\"odinger equation in the degenerate case

Changxing Miao·Xingdong Tang·Guixiang Xu
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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}.

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Changxing Miao, Xingdong Tang, Guixiang Xu. 2018-03-17. Instability of the solitary waves for the generalized derivative nonlinear Schr\"odinger equation in the degenerate case. https://doi.org/10.1016/j.jde.2023.02.061

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