Modulational Instability and Wave Dynamics in the Rotated Modified Gardner–Whitham Equation

Chaos, Solitons and Fractals. 2026. Vol. 213. No. 2. Article 119245.

Flamarion M. V., Pelinovsky E.

This article concerns the study of modulational instability in the rotated-modified Gardner–Whitham (rmGW) equation. This model incorporates both quadratic and cubic nonlinearities, similarly to the Gardner equation, while also retaining the fully dispersive character of the Whitham equation together with a large-scale dispersive term analogous to that in the Ostrovsky equation. Using a classical multiple-scale asymptotic expansion, we derive a cubic nonlinear Schrödinger equation governing the evolution of weakly modulated wave packets and analyze the corresponding modulational instability regime. Direct numerical simulations confirm the asymptotic predictions. In the defocusing regime, wave packets tend to disperse and spread out, whereas in the focusing regime they undergo compression and progressive steepening. Subharmonic perturbations of periodic wave trains are also considered to excite the Benjamin–Feir instability, with numerical growth rates showing good agreement with theoretical predictions at small amplitudes. In addition, the evolution of nearly peakon-type and thick solitary wave profiles is discussed. Regarding the last, an initial thick solitary wave undergoes a cascade of fission events, producing successive undular bores and
dispersive wavetrains before eventually breaking down into a conventional solitary wave that subsequently evolves into a localized wave packet.

Keywords: солитон, soliton