On the ground states of the Ostrovskyi equation and their stabilit

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.