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arXiv · 2004.10866

On the Barashenkov-Bogdan-Zhanlav solitons and their stability

Wen Feng·Milena Stanislavova·Atanas G. Stefanov
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Abstract

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

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Wen Feng, Milena Stanislavova, Atanas G. Stefanov. 2020-04-22. On the Barashenkov-Bogdan-Zhanlav solitons and their stability. https://doi.org/10.1016/j.chaos.2021.111467

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