Formation of wave packets in the Ostrovsky equation for both normal and anomalous dispersion

Abstract

It is well known that the Ostrovsky equation with normal dispersion does not support steady solitary waves. An initial Korteweg-de Vries solitary wave decays adiabatically through the radiation of long waves and is eventually replaced by an envelope solitary wave whose carrier wave and envelope move with different velocities (phase and group velocities correspondingly). Here, we examine the same initial condition for the Ostrovsky equation with anomalous dispersion, when the wave frequency increases with wavenumber in the limit of very short waves. The essential difference is that now there exists a steady solitary wave solution (Ostrovsky soliton), which in the small-Amplitude limit can be described asymptotically through the solitary wave solution of a nonlinear Schrödinger equation, based at that wavenumber where the phase and group velocities coincide. Long-Time numerical simulations show that the emergence of this steady envelope solitary wave is a very robust feature. The initial Korteweg- de Vries solitary wave transforms rapidly to this envelope solitary wave in a seemingly non-Adiabatic manner. The amplitude of the Ostrovsky soliton strongly correlates with the initial Korteweg-de Vries solitary wave.