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Spectral stability of constrained solitary waves for a generalized Ostrovsky equation

Part of: Incompressible inviscid fluids Partial differential equations Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  28 November 2024

Fangyu Han  and
Yuetian Gao
Fangyu Han
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China ([email protected])
Yuetian Gao
Affiliation:
School of Mathematics and Statistics, Fujian Normal University, Qishan Campus, Fuzhou 350117, People’s Republic of China ([email protected]) (corresponding author)
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Abstract

We consider the existence and stability of constrained solitary wave solutions to the generalized Ostrovsky equation

\begin{align*}\partial_x\left(\partial_t u+ \alpha\partial_x u+\partial_x(f(u))+\beta \partial_x^3u\right)=\gamma u,\quad \|u\|_{L^2}^2=\lambda \gt 0,\end{align*}
where the homogeneous nonlinearities $f(s)=\alpha_0|s|^p+\alpha_1|s|^{p-1}s$, with p > 1. If $\alpha_0,\alpha_1 \gt 0$, $\alpha\in\mathbb{R}$, and γ < 0 satisfying $\beta\gamma=-1$, we show that for $1 \lt p \lt 5$, there exists a constrained ground state traveling wave solution with travelling velocity $\omega \gt \alpha-2$. Furthermore, we obtain the exponential decay estimates and the weak non-degeneracy of the solution. Finally, we show that the solution is spectrally stable. This is a continuation of recent work [1] on existence and stability for a water wave model with non-homogeneous nonlinearities.

Keywords

solitary wavespectral stabilitythe generalized Ostrovsky equation

MSC classification

Primary: 35Q35: PDEs in connection with fluid mechanics
Secondary: 76B25: Solitary waves 35B35: Stability
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 32
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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