Instability of the solitary waves for the 1d NLS with an attrictive delta potential in the degenerate case
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| Format: | Preprint |
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2019
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| _version_ | 1866911850645422080 |
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| author | Tang, Xingdong Xu, Guixiang |
| author_facet | Tang, Xingdong Xu, Guixiang |
| contents | In this paper, we show the orbital instability of the solitary waves $Q_Ωe^{iΩt}$ of the 1d NLS with an attractive delta potential ($γ>0$)
\begin{equation*}
ıu_t+u_{xx}+γδu+\abs{u}^{p-1}u=0, \; p>5,
\end{equation*} where $Ω=Ω(p,γ)>\frac{γ^2}{4}$ is the critical oscillation number and determined by
\begin{equation*} \frac{p-5}{p-1} \int_{ \arctanh\sts{ \fracγ{2\sqrtΩ} } }^{+\infty} \sech^{\frac{4}{p-1}}\sts{y}\d y =
{ \fracγ{ 2\sqrtΩ } }\sts{ 1-\frac{γ^2}{4Ω} }^{-\frac{p-3}{p-1}} \Longleftrightarrow \mathbf{d}''(Ω) =0.
\end{equation*} The classical convex method and Grillakis-Shatah-Strauss's stability approach in \cite{A2009Stab, GSS1987JFA1} don't work in this degenerate case, and the argument here is motivated by those in \cite{CP2003CPAM, MM2001GAFA, M2012JFA, MTX2018, O2011JFA}. The main ingredients are to construct the unstable second order approximation near the solitary wave $Q_Ωe^{iΩt}$ on the level set $\Mcal(Q_Ω)$ accoding to the degenerate structure of the Hamiltonian and to construct the refined Virial identity to show the orbital instability of the solitary waves $Q_Ωe^{iΩt}$ in the energy space. Our result is the complement of the results in \cite{FOO2008AIHP} in the degenerate case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_11150 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Instability of the solitary waves for the 1d NLS with an attrictive delta potential in the degenerate case Tang, Xingdong Xu, Guixiang Analysis of PDEs Primary: 35L70, Secondary: 35Q55 In this paper, we show the orbital instability of the solitary waves $Q_Ωe^{iΩt}$ of the 1d NLS with an attractive delta potential ($γ>0$) \begin{equation*} ıu_t+u_{xx}+γδu+\abs{u}^{p-1}u=0, \; p>5, \end{equation*} where $Ω=Ω(p,γ)>\frac{γ^2}{4}$ is the critical oscillation number and determined by \begin{equation*} \frac{p-5}{p-1} \int_{ \arctanh\sts{ \fracγ{2\sqrtΩ} } }^{+\infty} \sech^{\frac{4}{p-1}}\sts{y}\d y = { \fracγ{ 2\sqrtΩ } }\sts{ 1-\frac{γ^2}{4Ω} }^{-\frac{p-3}{p-1}} \Longleftrightarrow \mathbf{d}''(Ω) =0. \end{equation*} The classical convex method and Grillakis-Shatah-Strauss's stability approach in \cite{A2009Stab, GSS1987JFA1} don't work in this degenerate case, and the argument here is motivated by those in \cite{CP2003CPAM, MM2001GAFA, M2012JFA, MTX2018, O2011JFA}. The main ingredients are to construct the unstable second order approximation near the solitary wave $Q_Ωe^{iΩt}$ on the level set $\Mcal(Q_Ω)$ accoding to the degenerate structure of the Hamiltonian and to construct the refined Virial identity to show the orbital instability of the solitary waves $Q_Ωe^{iΩt}$ in the energy space. Our result is the complement of the results in \cite{FOO2008AIHP} in the degenerate case. |
| title | Instability of the solitary waves for the 1d NLS with an attrictive delta potential in the degenerate case |
| topic | Analysis of PDEs Primary: 35L70, Secondary: 35Q55 |
| url | https://arxiv.org/abs/1910.11150 |