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.
I will talk about constructing ground traveling waves for these models as minimizers of the Hamiltonian among all waves with fixed L2 norm. The existence argument proceeds via the method of compensated compactness, but it requires surprisingly detailed Fourier analysis arguments to rule out the non-triviality of the limits of the minimizing sequences. We show that (at least almost all of) the waves are strongly spectrally stable, along with other properties: smoothness with respect to parameters, weak non-degeneracy of the waves etc.
In the case of quadratic nonlinearities, it is known that these waves are unique, by work of Zhang-Liu. Thus, our results imply in particular that (at least almost all of) the Zhang-Liu waves are spectrally stable. This is joint work with professor A. Stefanov.