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Transverse instability for a system of nonlinear Schrödinger equations

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  • 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 $\mathbb{R} \times \mathbb{T}_L$.
    Mathematics Subject Classification: Primary: 35B35, 35C08, 35Q55.


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