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

TL;DR

This paper investigates Hamiltonian Hopf bifurcations in nonlinear Schrödinger equations, demonstrating how these bifurcations lead to complex dynamics and chaos through a combination of analytical and numerical methods.

Contribution

It constructs localized potentials inducing HH bifurcations and reduces PDEs to ODEs, revealing chaotic behavior and bifurcation conditions.

Findings

PDE dynamics are well-approximated by the reduced ODE system

Both PDE and ODE exhibit Hamiltonian chaos

Conditions for HH bifurcation are derived and explained

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.