On (in)stability of discontinuous standing waves for the NLS with a delta-prime on star graphs
We investigate the existence and (in)stability of standing waves that are discontinuous at the vertex for the one-dimensional nonlinear Schrödinger equation (NLS) with a power nonlinearity, posed on a star graph consisting of a finite number $N$ of half-lines and endowed with a delta-prime interaction. Our non-variational approach characterizes all possible configurations of such standing wave profiles and shows, rather surprisingly, that the components of any discontinuous-at-the-vertex stationary solution split into two groups, each determined by one of exactly two distinct types of shifted soliton for the NLS on a half-line. More precisely, for every $N\geqq 2$ and every $n\in\{1,\dots,N-1\}$, there are two groups consisting of exactly $n$ and $N-n$ components, with each group sharing a common shift. The stability properties of these discontinuous-at-the-vertex stationary states are analyzed via the Grillakis-Shatah-Strauss framework and the Grillakis-Jones Instability Theorem, where non-standard techniques are required to establish the necessary spectral properties of the linearization operators, in particular their Morse and deficiency indices. Our approach yields a complete description of these indices. The results of this manuscript optimally extend previous findings in the literature concerning delta-prime type interactions on star graphs. Moreover, the methods developed here have the prospect of being adapted to study the stability of other discontinuous-at-the-vertex stationary solutions of the NLS on non-compact metric graphs, such as looping-edge graphs.