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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.