Interaction of sine-Gordon kinks with defects: The two-bounce resonance
Roy Goodman and Richard Haberman. Physica D: Nonlinear Phenomena, 2004.
Abstract
A model of soliton–defect interactions in the sine-Gordon equations is studied using singular perturbation theory. Melnikov theory is used to derive a critical velocity for strong interactions, which is shown to be exponentially small for weak defects. Matched asymptotic expansions for nearly heteroclinic orbits are constructed for the initial value problem, which are then used to derive analytical formulas for the locations of the well known two- and three-bounce resonance windows, as well as several other phenomena seen in numerical simulations.
Type
Publication
Physica D: Nonlinear Phenomena, 195(3-4), 303–323
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
Professor
Roy approached Rich with a problem that he was stuck on in 2002. Thus began a long-term friendship and collaboration. Rich passed away in December 2018. See a remembrance here.