Transverse instability for a system of nonlinear Schrödinger equations

Yohei Yamazaki

doi:10.3934/dcdsb.2014.19.565

Article Contents

2014, Volume 19, Issue 2: 565-588. Doi: 10.3934/dcdsb.2014.19.565

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Transverse instability for a system of nonlinear Schrödinger equations

Yohei Yamazaki^1,

1.
Department of Mathematics, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo, Kyoto 606-8502

Received: October 31, 2012

Revised: October 31, 2013

Published: February 2014

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Abstract

In this paper, we consider the transverse instability for a system of nonlinear Schrödinger equations on $\mathbb{R} \times \mathbb{T}_L $. Here, $\mathbb{T}_L$ means the torus with a $2\pi L$ period. It was shown by Colin-Ohta [11] that this system on $\mathbb{R}$ has a stable standing wave. In this paper, we regard this standing wave as the standing wave of this system on $\mathbb{R} \times \mathbb{T}_L$. Then, we show that there exists the critical period $L_{\omega}$ which is the boundary between the stability and the instability of the standing wave on $\mathbb{R} \times \mathbb{T}_L$.

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Mathematics Subject Classification: Primary: 35B35, 35C08, 35Q55.

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