Dynamical splitting conjecture for shifted states on star graphs

Let $K=1$ and consider the shifted state on a star graph with shift parameter $a<0$. The graph has one incoming edge and $N-1$ outgoing edges, meeting at the vertex $x=0$; a solitary wave is a localized travelling wave of the NLS flow.

Dynamical splitting conjecture. In the case $K=1$, the shifted state with $a<0$ leads to a solitary wave that moves towards the vertex point at $x=0$ along the only incoming edge, splits into $N-1$ solitary waves in the $N-1$ outgoing edges, which transform due to their spectral instability while moving outward from the vertex point at $x=0$.

This conjecture records the dynamical picture suggested by the momentum balance and travelling-wave calculations. The supplied text says that the validity of this picture is beyond the scope of the work and remains to be proved.