Interaction of interfacial waves with an external force: The Benjamin-Ono equation framework

Symmetry. 2023. Vol. 15. No. 8. Article 1478.

Flamarion M. V., Pelinovsky E.

This study aims to explore the complex interactions between an internal solitary wave and
an external force using the Benjamin-Ono equation as the theoretical framework. The investigation
encompasses both asymptotic and numerical approaches. By assuming a small amplitude for the
external force, we derive a dynamical system that describes the behavior of the solitary wave
amplitude and the position of its crest. Our findings reveal three distinct scenarios: (i) resonance
between the solitary wave and the external force, (ii) oscillatory motion with closed orbits, and
(iii) displacement from the initial position while maintaining the wave direction. However, through
numerical simulations, we observe a different relationship between the amplitude of the solitary wave
and its crest position. Specifically, for external forces of small amplitude, the simulations indicate
the presence of an unstable spiral pattern. Conversely, when subjected to external forces of larger
amplitudes, the solitary wave exhibits a stable spiral trajectory which resembles the classical damped
mass-spring system.

Keywords: Benjamin-Ono equation, уравнение Бенджамина-Оно