On the ground states of the Ostrovskyi equation and their stabilit

TL;DR

This paper rigorously constructs ground traveling wave solutions for the Ostrovskyi equation, demonstrating their existence, stability, and non-degeneracy through advanced Fourier analysis and variational methods.

Contribution

It introduces a novel variational approach to establish the existence and stability of ground states for the Ostrovskyi equation using compensated compactness.

Findings

Existence of ground traveling waves as Hamiltonian minimizers.

All ground states are weakly non-degenerate.

Ground states are spectrally stable.

Abstract

The Ostrovskyi (Ostrovskyi-Vakhnenko/short pulse) equations are ubiquitous models in mathematical physics. They describe water waves under the action of a Coriolis force as well as the amplitude of a "short" pulse in an optical fiber. In this paper, we rigorously construct ground traveling waves for these models as minimizers of the Hamiltonian functional for any fixed $L^2$ norm. The existence argument proceeds via the method of compensated compactness, but it requires surprisingly detailed Fourier analysis arguments to rule out the non-vanishing of the limits of the minimizing sequences. We show that all of these waves are weakly non-degenerate and spectrally stable.