Strong instability of standing waves with negative energy for double power nonlinear Schr\"odinger equations
Strong instability of standing waves with negative energy for double power nonlinear Schr\"odinger equations
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发表时间：
2018-06
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通讯作者：
Noriyoshi Fukaya;Masahito Ohta
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作者：
Noriyoshi Fukaya;Masahito Ohta
研究了具有双幂次非线性的$N$维非线性Schrödinger方程中基态驻波$e^{i\omega t}\phi_\omega(x)$的强不稳定性。一个是$L^2$ -亚临界，另一个是$L^2$ -超临界。Ohta和Yamaguchi（2015）证明了正能量驻波的强不稳定性。在本文中，我们改进了先前的结果，即证明了如果$\partial_\lambda^2S_\omega(\phi_\omega^\lambda)|_{\lambda=1}\le0$，驻波是强不稳定的，其中$S_\omega$为作用，$\phi_\omega^\lambda(x)\mathrel{\mathop:}=\lambda^{N/2}\phi_\omega(\lambda x)$为$L^2$不变标度。
We study the strong instability of ground-state standing waves $e^{i\omega t}\phi_\omega(x)$ for $N$-dimensional nonlinear Schr\"odinger equations with double power nonlinearity. One is $L^2$-subcritical, and the other is $L^2$-supercritical. The strong instability of standing waves with positive energy was proven by Ohta and Yamaguchi (2015). In this paper, we improve the previous result, that is, we prove that if $\partial_\lambda^2S_\omega(\phi_\omega^\lambda)|_{\lambda=1}\le0$, the standing wave is strongly unstable, where $S_\omega$ is the action, and $\phi_\omega^\lambda(x)\mathrel{\mathop:}=\lambda^{N/2}\phi_\omega(\lambda x)$ is the $L^2$-invariant scaling.