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2003 Instability of standing waves for nonlinear Schrödinger equations with potentials
Reika Fukuizumi, Masahito Ohta
Differential Integral Equations 16(6): 691-706 (2003).

Abstract

We study the instability of standing waves $e^{i \omega t}\phi_{\omega}(x)$ for a nonlinear Schrödinger equation with an attractive power nonlinearity $|u|^{p-1}u$ and a potential $V(x)$ in ${\mathbb{R}}^n$. Here, $\omega>0$ and $\phi_{\omega}(x)$ is a minimal action solution of the stationary problem. Under suitable assumptions on $V(x)$, we show that if $p>1+4/n$, $e^{i \omega t}\phi_{\omega}(x)$ is unstable for sufficiently large $\omega$. For example, our theorem covers a harmonic potential $V(x)=|x|^2$, to which the arguments in the previous papers [2], [14], and [19] are not directly applicable. As another application, we also prove a similar result for a nonlinear Schrödinger equation with a constant magnetic field.

Citation

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Reika Fukuizumi. Masahito Ohta. "Instability of standing waves for nonlinear Schrödinger equations with potentials." Differential Integral Equations 16 (6) 691 - 706, 2003.

Information

Published: 2003
First available in Project Euclid: 21 December 2012

zbMATH: 1031.35131
MathSciNet: MR1973275

Subjects:
Primary: 35Q55
Secondary: 35R25

Rights: Copyright © 2003 Khayyam Publishing, Inc.

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Vol.16 • No. 6 • 2003
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