Analytical Proof of Space-Time Chaos in Ginzburg-Landau Equations

We prove that the attractor of the 1D quintic complex Ginzburg-Landau equation with a broken phase symmetry has strictly positive space-time entropy for an open set of parameter values. The result is obtained by studying chaotic oscillations in grids of weakly interacting solitons in a class of Ginzburg-Landau type equations. We provide an analytic proof for the existence of two-soliton configurations with chaotic temporal behavior, and construct solutions which are closed to a grid of such chaotic soliton pairs, with every pair in the grid well spatially separated from the neighboring ones for all time. The temporal evolution of the well-separated multi-soliton structures is described by a weakly coupled lattice dynamical system (LDS) for the coordinates and phases of the solitons. We develop a version of normal hyperbolicity theory for the weakly coupled LDSs with continuous time and establish for them the existence of space-time chaotic patterns similar to the Sinai-Bunimovich chaos in discrete-time LDSs. While the LDS part of the theory may be of independent interest, the main difficulty addressed in the paper concerns with lifting the space-time chaotic solutions of the LDS back to the initial PDE. The equations we consider here are space-time autonomous, i.e. we impose no spatial or temporal modulation which could prevent the individual solitons in the grid from drifting towards each other and destroying the well-separated grid structure in a finite time. We however manage to show that the set of space-time chaotic solutions for which the random soliton drift is arrested is large enough, so the corresponding space-time entropy is strictly positive

Chaos-driven dynamics in spin-orbit coupled atomic gases

The dynamics, appearing after a quantum quench, of a trapped, spin-orbit coupled, dilute atomic gas is studied. The characteristics of the evolution is greatly influenced by the symmetries of the system, and we especially compare evolution for an isotropic Rashba coupling and for an anisotropic spin-orbit coupling. As we make the spin-orbit coupling anisotropic, we break the rotational symmetry and the underlying classical model becomes chaotic; the quantum dynamics is affected accordingly. Within experimentally relevant time-scales and parameters, the system thermalizes in a quantum sense. The corresponding equilibration time is found to agree with the Ehrenfest time, i.e. we numerically verify a ~log(1/h) scaling. Upon thermalization, we find the equilibrated distributions show examples of quantum scars distinguished by accumulation of atomic density for certain energies. At shorter time-scales we discuss non-adiabatic effects deriving from the spin-orbit coupled induced Dirac point. In the vicinity of the Dirac point, spin fluctuations are large and, even at short times, a semi-classical analysis fails.Comment: 11 pages, 10 figure

Tunable transport with broken space-time symmetries

Transport properties of particles and waves in spatially periodic structures that are driven by external time-dependent forces manifestly depend on the space-time symmetries of the corresponding equations of motion. A systematic analysis of these symmetries uncovers the conditions necessary for obtaining directed transport. In this work we give a unified introduction into the symmetry analysis and demonstrate its action on the motion in one-dimensional periodic, both in time and space, potentials. We further generalize the analysis to quasi-periodic drivings, higher space dimensions, and quantum dynamics. Recent experimental results on the transport of cold and ultracold atomic ensembles in ac-driven optical potentials are reviewed as illustrations of theoretical considerations.Comment: Phys. Rep., in pres

A new deterministic model for chaotic reversals

We present a new chaotic system of three coupled ordinary differential equations, limited to quadratic nonlinear terms. A wide variety of dynamical regimes are reported. For some parameters, chaotic reversals of the amplitudes are produced by crisis-induced intermittency, following a mechanism different from what is generally observed in similar deterministic models. Despite its simplicity, this system therefore generates a rich dynamics, able to model more complex physical systems. In particular, a comparison with reversals of the magnetic field of the Earth shows a surprisingly good agreement, and highlights the relevance of deterministic chaos to describe geomagnetic field dynamics.Comment: 12 pages, 14 figures, accepted in EPJ