A Mechanical Analog of the Two-bounce Resonance of Solitary Waves: Modeling and Experiment

Abstract

We describe a simple mechanical system, a ball rolling along a specially-designed landscape, that mimics the dynamics of a well known phenomenon, the two-bounce resonance of solitary wave collisions, that has been seen in countless numerical simulations but never in the laboratory. We provide a brief history of the solitary wave problem, stressing the fundamental role collective-coordinate models played in understanding this phenomenon. We derive the equations governing the motion of a point particle confined to such a surface and then design a surface on which to roll the ball, such that its motion will evolve under the same equations that approximately govern solitary wave collisions. We report on physical experiments, carried out in an undergraduate applied mathematics course, that seem to verify one aspect of chaotic scattering, the so-called two-bounce resonance.

9 pages, 8 figures, link to one youtube video

References in corpus (4)

• Interaction of topological solitons with defects: using a nontrivial metric
• Chaotic scattering in solitary wave interactions: A singular iterated-map description
• Scattering of Sine-Gordon kinks on potential wells
• Chaotic Scattering and the $n$-bounce Resonance in Solitary Wave Interactions

Cited by in corpus (6)

• Collective Coordinates in One-Dimensional Soliton Models Revisited
• Kink-Antikink Collisions and Multi-Bounce Resonance Windows in Higher-Order Field Theories
• Kink Dynamics in a Parametric ${\varphi }^6$ System: A Model With Controllably Many Internal Modes
• Fractal structure of the soliton scattering for the graphene superlattice equation
• Crises and chaotic scattering in hydrodynamic pilot-wave experiments
• Multi-bounce resonances in the interaction of walking droplets