On the Barashenkov-Bogdan-Zhanlav solitons and their stability

The Barashenkov-Bogdan-Zhanlav solitons $u_\pm$ for the forced NLS/Lugiato-Lefever model on line are considered. While instability of $u_+$ was established in original paper, \cite{B1}, analogous question $u_-$ only considered heuristically and numerically. We rigorously analyze stability various regime parameters. In particular, we show that is spectrally stable small pump strength $h$. Moreover, remains until a pair neutral eigenvalues negative Krein signature hits another eigenvalues, which has emanated from edge continuous spectrum, \cite{B1, BBK, ABP}. After collision, an conjectured numerically observed previous works, \cite{B1}.