Hamiltonian Hopf bifurcations and dynamics of NLS/GP standing-wave modes

Roy Goodman. Journal of Physics A: Mathematical and Theoretical, 2011.
Jan 1, 2011·
Roy Goodman
Roy Goodman
· 1 min read
Abstract
We examine the dynamics of solutions to nonlinear Schrödinger/Gross– Pitaevskii equations that arise due to semisimple indefinite Hamiltonian Hopf bifurcations—the collision of pairs of eigenvalues on the imaginary axis. We construct localized potentials for this model which lead to such bifurcations in a predictable manner. We perform a formal reduction from the partial differential equations to a small system of ordinary differential equations. We analyze the equations to derive conditions for this bifurcation and use averaging to explain certain features of the dynamics that we observe numerically. A series of careful numerical experiments are used to demonstrate the phenomenon and the relations between the full system and the derived approximations.
Type
Publication
Journal of Physics A: Mathematical and Theoretical, 44, 425101
publication published

Erratum: At the end of section 2.4.3, top of page 10, I state that “a straightforward calculation” shows that the HH bifurcation studied by Johansson for the periodic NLS trimer to come from a semisimple -1:1 resonance. In fact, it is the generic non-semisimple bifurcation.

Bifurcation Nonlinear Schrödinger equation
Roy Goodman
Authors
Roy Goodman
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.