A Mechanical Analog of the Two-Bounce Resonance of Solitary Waves: Modeling and Experiment
Roy Goodman, Amin Rahman, Michael Bellanich, and Catherine Morrison. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2015.
Jan 1, 2015·
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1 min read
Roy Goodman
Amin Rahman
Michael Bellanich
Catherine Morrison
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.
Type
Publication
Chaos: An Interdisciplinary Journal of Nonlinear Science, 25, 043109
For more about NJIT’s applied math capstone class, see here.
Authors
Professor, Associate Chair for Graduate Studies, Department of Mathematical Sciences
I am a professor in the Department of Mathematical Sciences at NJIT. My research applies methods of dynamical systems to (conservative, dispersive) nonlinear wave equations arising, for example, in optics and fluid mechanics. I am interested in dynamical and bifurcation phenomena, and I use techniques from Hamiltonian mechanics and perturbation theory, as well as extensive computations. My current research in vortex dynamics is supported by the NSF under DMS-2206016.
I led the effort to start the SIAM New York-New Jersey-Pennsylvania Section (SIAM NNP) and was its first president in 2023-2024. I have handed over the reins to a new leadership group.
Authors
Senior Scientist
Amin did his undergrad and Ph.D. at NJIT. After he took the capstone class with me, he and two other students worked to turn that project into a research paper.