ORBITAL INSTABILITY OF STANDING WAVES FOR NLS EQUATION ON STAR GRAPHS

We consider a nonlinear Schrodinger (NLS) equation with any positive power nonlinearity on a star graph Gamma (N half-lines glued at the common vertex) with a delta interaction at the vertex. The strength of the interaction is defined by a fixed value alpha is an element of R. In the recent works of Adami et al., it was shown that for alpha not equal 0 the NLS equation on Gamma admits the unique symmetric (with respect to permutation of edges) standing wave and that all other possible standing waves are nonsymmetric. Also, it was proved for alpha < 0 that in the NLS equation with a subcritical power-type nonlinearity, the unique symmetric standing wave is orbitally stable. In this paper, we analyze stability of standing waves for both alpha < 0 and alpha > 0. By extending the Sturm theory to Schrodinger operators on the star graph, we give the explicit count of the Morse and degeneracy indices for each standing wave. For alpha < 0, we prove that all nonsymmetric standing waves in the NLS equation with any positive power nonlinearity are orbitally unstable. For alpha > 0, we prove the orbital instability of all standing waves.