Transverse instability for a system of nonlinear Schrödinger equations

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