STABILITY OF LINE STANDING WAVES NEAR THE BIFURCATION POINT FOR NONLINEAR SCHRODINGER EQUATIONS

In this paper we consider the transverse instability for a nonlinear Schrodinger equation with power nonlinearity on R x T-L, where 2 pi L is the period of the torus T-L. There exists a critical period 2 pi L-omega,L-p such that the line standing wave is stable for L < L-omega,L-p and the line standing wave is unstable for L > L-omega,L-p. Here we farther study the bifurcation from the boundary L = L-omega,L-p between the stability and the instability for line standing waves of the nonlinear Schrodinger 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].