Long-time solutions of the Ostrovsky equation

The Ostrovsky equation is a modification of the Korteweg-de Vries equation which takes account of the effects of background rotation. It is well known that then the usual Korteweg-de Vries solitary wave decays and is replaced by radiating inertia-gravity waves. Here we show through numerical simulations that after a long-time a localized wave packet emerges as a persistent and dominant feature. The wavenumber of the carrier wave is associated with that critical wavenumber where the underlying group velocity is a a minimum (in absolute value). Based on this feature, we construct a weakly nonlinear theory leading to a higher-order nonlinear Schrödinger equation in an attempt to describe the numerically found wave packets.