Instability of the solitary waves for the generalized derivative nonlinear Schrödinger equation in the degenerate case
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2018
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| _version_ | 1866929325716013056 |
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| author | Miao, Changxing Tang, Xingdong Xu, Guixiang |
| author_facet | Miao, Changxing Tang, Xingdong Xu, Guixiang |
| contents | 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1803_06451 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Instability of the solitary waves for the generalized derivative nonlinear Schrödinger equation in the degenerate case Miao, Changxing Tang, Xingdong Xu, Guixiang Analysis of PDEs 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}. |
| title | Instability of the solitary waves for the generalized derivative nonlinear Schrödinger equation in the degenerate case |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/1803.06451 |