Interaction of Interfacial Waves with an External Force: The Benjamin-Ono Equation Framework

Abstract

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.