Wave dynamics within the Whitham-Ostrovsky equation

Nonlinear Dynamics. 2026. Vol. 114. Article 784.

Flamarion M. V., Pelinovsky E.

In this article, we investigate wave packet and solitary wave dynamics in the Whitham–Ostrovsky (WO) equation. By means of a multiple-scales expansion, we formally derive a nonlinear Schrödinger (NLS) equation governing the envelope evolution.The corresponding modulational stability diagram is then obtained using the Lighthill criterion. We show that sufficiently large values of the low-frequency dispersive term render plane-wave solutions modulationally unstable. Direct numerical simulations confirm that, within the unstable region, wave packets undergo a pronounced compression, consistent with the self-focusing mechanism of the focusing NLS equation. In contrast, in the modulationally stable region, the wave packet progressively broadens in space while its peak amplitude decreases, as the wave energy is redistributed over an increasingly wider spatial interval. We further examine how solitary-wave solutions of the Whitham equation are modified within the WO framework, where they evolve into localized wave packets due to the presence of the rotating term. In addition, we investigate the dynamics of solitary waves in the anomalous dispersion regime. These solutions are computed numerically and evolved under the full time-dependent equation, revealing that their interactions are inelastic, with noticeable generation of dispersive radiation and an increase in the amplitude of the larger solitary wave. Although such interactions could, in
principle, promote the emergence of a “soliton champion” or even a freak-wave–type structure after repeated collisions, the progressive steepening of the solitary waves alters this scenario. Numerical simulations indicate that, instead of forming a persistent dominant soliton, the wave profile continues to sharpen until the onset of wave breaking occurs.

Keywords: нелинейная динамика, nonlinear dynamics, Whitham equation, уравнение Уизема, Ostrovsky equation, уравнение Островского