1. Hamiltonian Hopf bifurcations and dynamics of NLS/GP standing-wave modes
- Author
-
Roy H. Goodman
- Subjects
Statistics and Probability ,Physics ,Partial differential equation ,FOS: Physical sciences ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Pattern Formation and Solitons (nlin.PS) ,Nonlinear Sciences - Chaotic Dynamics ,Nonlinear Sciences - Pattern Formation and Solitons ,70H08, 35J10, 78A60 ,Standing wave ,Nonlinear system ,symbols.namesake ,Classical mechanics ,Modeling and Simulation ,Ordinary differential equation ,symbols ,Chaotic Dynamics (nlin.CD) ,Hamiltonian (quantum mechanics) ,Mathematical Physics ,Eigenvalues and eigenvectors ,Bifurcation ,Schrödinger's cat - Abstract
We examine the dynamics of solutions to nonlinear Schrodinger/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., Comment: submitted to J. Phys. A., 32 pages, many figures
- Published
- 2011
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