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.

Decay of Benjamin–Ono solitons under the influence of dissipation

“…In this case the initial BO soliton does not decay, but gradually evolves into another soliton of a higher amplitude and zero mean value. A similar situation occurs for KdV solitons which gradually transform into Ostrovsky solitons under the influence of large scale dispersion [14]. In Fig.…”

Whitham modulation theory for the Ostrovsky equation

“…The variables and analysis of §3 give a context for discussing the emergence of the wavepacket solutions above from KdV soliton initial conditions under weak rotation as observed by Grimshaw & Helfrich [8], Grimshaw et al [7] and Whitfield & Johnson [9]. In terms of the variables (2.4), equation (2.1) has the inner solitary wave solution …”

“…In the strong-rotation limit ( 1), these solutions originate from the point in wavenumber space where the linear phase and group velocities are equal [7], but as the strength of rotation is decreased the wavepacket structure of this family of solutions is eventually replaced by a soliton solution (figure 2a) that is described asymptotically by a KdV soliton with pedestal of order 1 [15]. Section 4 describes how these single-peak solitary wave solutions arise naturally from the evolution of an initial KdV soliton in the presence of weak rotation.…”

“…The effect of adding the non-local integral term to the right-hand side of the KdV equation is to introduce 'large-scale' dispersion into the system. For oceanic waves, the coefficient representing the strength of rotation is usually positive, γ > 0, and this parameter regime has been described as the Ostrovsky equation with normal dispersion [7]. The opposite case, when γ < 0, arises in strongly sheared ocean flows and acoustic problems [7] and is described as anomalous dispersion.…”

“…In the strong-rotation limit (γ 1), it has been shown that the Ostrovsky equation supports localized wavepacket solutions for both cases of dispersion and several recent articles [7][8][9][10] have examined the dynamics of these wavepackets. In the weak-rotation limit (γ 1), wavepackets have been observed for normal dispersion only.…”