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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Transverse instability for a system of nonlinear Schrödinger equations

Pages: 565 - 588, Volume 19, Issue 2, March 2014      doi:10.3934/dcdsb.2014.19.565

 
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Yohei Yamazaki - Department of Mathematics, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo, Kyoto 606-8502, Japan (email)

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 $\mathbb{R} \times \mathbb{T}_L$.

Keywords:  Stability, standing wave, Schrödinger equation.
Mathematics Subject Classification:  Primary: 35B35, 35C08, 35Q55.

Received: November 2012;      Revised: November 2013;      Available Online: February 2014.

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