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arXiv · 2403.10685

Orbital Stability of Smooth Solitary Waves for the Novikov Equation

Brett Ehrman·Mathew A. Johnson·Stéphane Lafortune
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Abstract

We study the orbital stability of smooth solitary wave solutions of the Novikov equation, which is a Camassa-Holm type equation with cubic nonlinearities. These solitary waves are shown to exist as a one-parameter family (up to spatial translations) parameterized by their asymptotic endstate, and are encoded as critical points of a particular action functional. As an important step in our analysis we must study the spectrum the Hessian of this action functional, which turns out to be a nonlocal integro-differential operator acting on $L^2(\mathbb{R})$. We provide a combination of analytical and numerical evidence that the necessary spectral hypotheses always holds for the Novikov equation. Together with a detailed study of the associated Vakhitov-Kolokolov condition, our analysis indicates that all smooth solitary wave solutions of the Novikov equation are nonlinearly orbitally stable.

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Brett Ehrman, Mathew A. Johnson, Stéphane Lafortune. 2024-03-15. Orbital Stability of Smooth Solitary Waves for the Novikov Equation. https://arxiv.org/abs/2403.10685

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