On the ground states of the Ostrovskyi equation and their stability

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 2 norm. The existence argument proceeds via the method of compensated compactness, but it requires surprisingly detailed Fourier analysis arguments to rule out the nonvanishing of the limits of the minimizing sequences. We show that all of these waves are weakly nondegenerate and spectrally stable. K E Y W O R D S ground states, Ostrovskyi equation, stability 2 0 1 0 M A

Existence, regularity and symmetry of periodic traveling waves for Gardner–Ostrovsky type equations

“…Local well-posedness for the modified Ostrovsky equation is confirmed in Sobolev spaces with regularity index s ≥ 1 4 , while ill-posedness holds for s < 1 4 (see [44,45]). Solitary waves are constructed via variational arguments as ground state solutions in H 1 (R)-based energy spaces (see [15,31]) or H 2 (R)-based energy spaces (see [38]), or by Fenichel singular perturbation theory [10] with exponential decay. The stability is also considered in related function spaces in the these results (the stability of solitary solutions in H 1 (R)-based spaces is also studied in [30] for generalized Ostrovsky equations with homogeneous nonlinearities).…”