Interactions of solitons with an external force field: Exploring the Schamel equation framework

This study aims to investigate the interactions of solitons with an external force within the framework of the Schamel equation, both asymptotically and numerically. By utilizing asymptotic expansions, we demonstrate that the soliton interaction can be approximated by a dynamical system that involves the soliton amplitude and its crest position. To solve the Schamel equation, we employ a pseudospectral method and compare the obtained results with those predicted by the asymptotic theory. The asymptotic theory predicts that the soliton interaction can be classified into three categories: (i) steady interaction occurs when the crest of the soliton and the crest of the external force are in phase, (ii) oscillatory behavior arises when the soliton speed and the external force speed are close to resonance, causing the soliton to bounce back and forth near its initial position, and (iii) non-reversible motion occurs when the soliton moves away from its initial position without changing its direction. However, the numerical results indicate the presence of an unstable spiral pattern.