Stability of line standing waves near the bifurcation point for nonlinear Schrödinger equations

In this paper we consider the transverse instability for a nonlinear Schrödinger equation with power nonlinearity on R × T_L, where 2πL is the period of the torus T_L. There exists a critical period 2πL_{ω, p} such that the line standing wave is stable for L < L_{ω, p} and the line standing wave is unstable for L > L_{ω, p}. Here we farther study the bifurcation from the boundary L = L_{ω, p} between the stability and the instability for line standing waves of the nonlinear Schrödinger equation. We show the stability for the branch bifurcating from the line standing waves by applying the argument in Kirr, Kevrekidis and Pelinovsky [16] and the method in Grillakis, Shatah and Strauss [12]. However, at the bifurcation point, the linearized operator around the bifurcation point is degenerate. To prove the stability for the bifurcation point, we apply the argument in Maeda [18].