E. Shlizerman and V. Rom-Kedar, Three types of chaos in the forced nonlinear Schrödinger equation
Physical Review Letters, 96, 024104 (2006)
Abstract
Three different types of chaotic behavior and instabilities (homoclinic chaos, hyperbolic resonance and parabolic
resonance) in Hamiltonian perturbations of the nonlinear Schrödinger (NLS) are described. The analysis is performed
on a truncated model using a novel framework in which a hierarchy of bifurcations is constructed. Then, it is
demonstrated numerically that the forced NLS equation exhibits analogous types of chaotic phenomena. The study reveals
that an adjustment of the forcing frequency sets the behavior near the plane wave solution to one of the three
different types of chaos for any periodic box length.
Results
We provide a new classification of chaotic orbits in the perturbed, undamped two-mode model and reveal a
new type of chaotic behavior: parabolic resonance. Moreover, we suggest that in some phase-space regimes
there exists an analogous classification of the chaotic behavior of the forced NLS. Bellow we present excitations in the parabolic resonant regime.
Evolution of parabolic resonance regime for the forced periodic NLS pde. The initial condition is very close to the plane wave (Click on the picture to start the evolution).
Evolution of parabolic resonance regime for the forced periodic NLS pde for a small interval of time. The red color denotes the excited mode centered in the middle of the box and the green color stands for excited modes centered on the side of the box. The initial condition is very close to the plane wave (Click on the picture to start the evolution).
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