Roy Goodman | New Jersey Institute of Technology (original) (raw)

Papers by Roy Goodman

SIAM Journal on Applied Dynamical Systems, 2015

We study the motion of a vortex dipole in a Bose-Einstein condensate confined to an anisotropic t... more We study the motion of a vortex dipole in a Bose-Einstein condensate confined to an anisotropic trap. We focus on a system of ordinary differential equations describing the vortices' motion, which is in turn a reduced model of the Gross-Pitaevskii equation describing the condensate's motion. Using a sequence of canonical changes of variables, we reduce the dimension and simplify the equations of motion. We uncover two interesting regimes. Near a family of periodic orbits known as guiding centers, we find that the dynamics is essentially that of a pendulum coupled to a linear oscillator, leading to stochastic reversals in the overall direction of rotation of the dipole. Near the separatrix orbit in the isotropic system, we find other families of periodic, quasi-periodic, and chaotic trajectories. In a neighborhood of the guiding center orbits, we derive an explicit iterated map that simplifies the problem further. Numerical calculations are used to illustrate the phenomena discovered through the analysis. Using the results from the reduced system we are able to construct complex periodic orbits in the original, partial differential equation, mean-field model for Bose-Einstein condensates, which corroborates the phenomenology observed in the reduced dynamical equations.

We examine the dynamics of solutions to nonlinear Schrödinger/Gross-Pitaevskii equations that ari... more We examine the dynamics of solutions to nonlinear Schrödinger/Gross-Pitaevskii equations that arise due to Hamiltonian Hopf (HH) bifurcations-the collision of pairs of eigenvalues on the imaginary axis. To this end, we use inverse scattering to construct localized potentials for this model which lead to HH bifurcations in a predictable manner. We perform a formal reduction from the partial differential equations (PDE) to a small system of ordinary differential equations (ODE). We show numerically that the behavior of the PDE is well-approximated by that of the ODE and that both display Hamiltonian chaos. We analyze the ODE to derive conditions for the HH bifurcation and use averaging to explain certain features of the dynamics that we observe numerically.

Physical Review Letters, 2007

We present a new and complete analysis of the n-bounce resonance and chaotic scattering in solita... more We present a new and complete analysis of the n-bounce resonance and chaotic scattering in solitary wave collisions. In these phenomena, the speed at which a wave exits a collision depends in a complicated fractal way on its input speed. We present a new asymptotic analysis of collectivecoordinate ODEs, reduced models that reproduce the dynamics of these systems. We reduce the ODEs to discrete-time iterated separatrix maps and obtain new quantitative results unraveling the fractal structure of the scattering behavior. These phenomena have been observed repeatedly in many solitary-wave systems over 25 years.

Physical Review E, 2005

We consider the interactions of two identical, orthogonally polarized vector solitons in a nonlin... more We consider the interactions of two identical, orthogonally polarized vector solitons in a nonlinear optical fiber with two polarization directions, described by a coupled pair of nonlinear Schrödinger equations. We study a low-dimensional model system of Hamiltonian ordinary differential equations (ODEs) derived by Ueda and Kath and also studied by Tan and Yang. We derive a further simplified model which has similar dynamics but is more amenable to analysis. Sufficiently fast solitons move by each other without much interaction, but below a critical velocity the solitons may be captured.

Journal of Physics A: Mathematical and Theoretical, 2011

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2008

We derive a family of singular iterated maps-closely related to Poincaré maps-that describe chaot... more We derive a family of singular iterated maps-closely related to Poincaré maps-that describe chaotic interactions between colliding solitary waves. The chaotic behavior of such solitary wave collisions depends on the transfer of energy to a secondary mode of oscillation, often an internal mode of the pulse. Unlike previous analyses, this map allows one to understand the interactions in the case when this mode is excited prior to the first collision. The map is derived using Melnikov integrals and matched asymptotic expansions and generalizes a "multi-pulse" Melnikov integral and allows one to find not only multipulse heteroclinic orbits, but exotic periodic orbits. The family of maps derived exhibits singular behavior, including regions of infinite winding. This problem is shown to be a singular version of the conservative Ikeda map from laser physics and connections are made with problems from celestial mechanics and fluid mechanics.

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2015

We describe a simple mechanical system, a ball rolling along a specially-designed landscape, that... more 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.

Contemporary Mathematics, 2005

... These studies are beyond the scope of this paper and will be the subject of future publicatio... more ... These studies are beyond the scope of this paper and will be the subject of future publications. Software that imple-ments the neural network method is available at http://sprott.physics.wisc.edu/chaos/maus/Lagspace.htm References ...

$Research paper thumbnail of Self-trapping and Josephson tunneling solutions to the nonlinear Schr\" odinger/Gross-Pitaevskii Equation$

We study the long-time behavior of solutions to the nonlinear Schrödinger / Gross-Pitaevskii equa... more We study the long-time behavior of solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation (NLS/GP) with a symmetric double-well potential, continuing work of Marzuola and Weinstein . NLS/GP governs nearly-monochromatic guided optical beams in weakly coupled waveguides with both linear and nonlinear (Kerr) refractive indices and zero absorption. The optical power (L 2 norm) is conserved with propagation distance. At low optical power, the beam energy executes beating oscillations between the two waveguides. There is an optical power threshold above which the set of guided mode solutions splits into two families of solutions. One type of solution corresponds to an optical beam which is concentrated in either waveguide, but not both. Solutions in the second family undergo tunneling oscillations between the two waveguides. NLS/GP can also model the behavior of Bose-Einstein condensates. A finite dimensional reduction (system of ODEs) well-approximates the PDE dynamics on long time scales. In particular, we derive this reduction, find a class of exact solutions and prove the very long-time shadowing of these solutions by applying the approach of .