Dissipative Lattice Research Articles

Compact uniform attractors for dissipative lattice dynamical systems with delays

In this paper, weconsider the long time behavior of solutions for dissipative latticedynamical systems with delays. We first prove a sufficient andnecessary condition for the existence of a compact uniform attractorfor the family of processes corresponding to the lattice dynamicalsystems with delays. Then we apply this result to prove theexistence of a compact uniform attractor for the process associatedto the retarded lattice Zakharov equations. As a consequence, someresults for the non-delay lattice dynamical systems are deduced asparticular cases.

Dissipative dynamics of matter-wave solitons in a nonlinear optical lattice

Dynamics and stability of solitons in two-dimensional (2D) Bose-Einstein condensates (BEC), with one-dimensional (1D) conservative plus dissipative nonlinear optical lattices, are investigated. In the case of focusing media (with attractive atomic systems), the collapse of the wave packet is arrested by the dissipative periodic nonlinearity. The adiabatic variation of the background scattering length leads to metastable matter-wave solitons. When the atom feeding mechanism is used, a dissipative soliton can exist in focusing 2D media with 1D periodic nonlinearity. In the defocusing media (repulsive BEC case) with harmonic trap in one direction and nonlinear optical lattice in the other direction, the stable soliton can exist. Variational approach simulations are confirmed by full numerical results for the 2D Gross-Pitaevskii equation.

Compensation of the influence of loss for a spatial soliton in a dissipative modulated Bessel optical lattice

Propagation of a spatial soliton in a dissipative modulated Bessel optical lattice is investigated, both analytically and numerically. The dynamic evolution equations for beam width, amplitude, and curvature wavefront are obtained by a variational approach. It is shown that by properly increasing the modulation depth of refractive index of the optical lattice, the loss effect can be compensated exactly to fulfill a stable spatial soliton propagation.

Compact uniform attractors for dissipative non-autonomous lattice dynamical systems

This paper discusses the long time behavior of solutions for dissipative non-autonomous lattice dynamical systems. We first prove some sufficient and necessary conditions for the existence of a compact uniform attractor for the family of processes defined on a Hilbert space of infinite sequences, and then give an upper bound of the Kolmogorov $\varepsilon$-entropy for the uniform attractor. As an application, we consider the dissipative non-autonomous lattice Zakharov equations.

Kolmogorov’s ε-Entropy of Bounded Sets in Discrete Spaces and Attractors of Dissipative Lattice Systems

We obtain an estimate of the upper bound for Kolmogorov’s e-entropy for the bounded sets with small “tail” in discrete spaces, then we present a sufficient condition for the existence of a global attractor for dissipative lattice systems in a reflexive Banach discrete space and establish an upper bound of Kolmogorov’s e-entropy of the global attractor for lattice systems.

Nonlinear excitations and electric transport in dissipative Morse-Toda lattices

We investigate the onset and maintenance of nonlinear soliton-like excitations in chains of atoms with Morse interactions at rather high densities, where the exponential repulsion dominates. First we discuss the atomic interactions and approximate the Morse potential by an effective Toda potential with adapted density-dependent parameters. Then we study several mechanisms to generate and stabilize the soliton-like excitations: (i) External forcing: we shake the masses periodically, mimicking a piezoelectric-like excitation, and delay subsequent damping by thermal excitation; (ii) heating, quenching and active friction: we heat up the system to a relatively high temperature Gaussian distribution, then quench to a low temperature, and subsequently stabilize by active friction. Finally, we assume that the atoms in the chain are ionized with free electrons able to move along the lattice. We show that the nonlinear soliton-like excitations running on the chain interact with the electrons. They influence their motion in the presence of an external field creating dynamic bound states (“solectrons”, etc.). We show that these bound states can move very fast and create extra current. The soliton-induced contribution to the current is constant, field-independent for a significant range of values when approaching the zero-field value.

A nonadiabatic semi-classical method for dynamics of atoms in optical lattices

We develop a semi-classical method to simulate the motion of atoms in a dissipative optical lattice. Our method treats the internal states of the atom quantum mechanically, including all nonadiabatic couplings, while position and momentum are treated as classical variables. We test our method in the one-dimensional case. Excellent agreement with fully quantum mechanical simulations is found. Our results are much more accurate than those of earlier semi-classical methods based on the adiabatic approximation.

Asymptotic synchronization in n-dimensional dissipative lattices of coupled oscillators with Nuemann boundary condition

Asymptotic synchronization in n-dimensional dissipative lattice of coupled oscillators with Neumann boundary condition was considered by decomposing phase space into two orthogonal subspaces.

Tunable Tsallis Distributions in Dissipative Optical Lattices

We demonstrated experimentally that the momentum distribution of cold atoms in dissipative optical lattices is a Tsallis distribution. The parameters of the distribution can be continuously varied by changing the parameters of the optical potential. In particular, by changing the depth of the optical lattice, it is possible to change the momentum distribution from Gaussian, at deep potentials, to a power-law tail distribution at shallow optical potentials.

Attractors for partly dissipative lattice dynamic systems in weighted spaces

In this paper, we study the asymptotic behavior of solutions for the partly dissipative lattice dynamical systems in weighted spaces. We first establish the dynamic systems on infinite lattice, and then prove the existence of the global attractor in weighted spaces by the asymptotic compactness of the solutions. It is shown that the global attractors contain traveling waves. The upper semicontinuity of the global attractor is also considered by finite-dimensional approximations of attractors for the lattice systems.

Asymptotic synchronization in n-dimensional second order dissipative lattices of coupled oscillators

By introducing a new norm which is equivalent to the usual norm in the phase space, we prove that for n-dimensional second order dissipative lattices of coupled oscillators with external periodic forces under Dirichlet, Neumann and periodic boundary conditions, if the system is bounded dissipative and the coupled coefficients are both large enough, the asymptotic synchronization will occur. And we give a concrete bounded dissipative second order lattices system. Our results show that the bounds of the difference between the components of any solution are directly proportional to m n / 2 and inversely proportional to the coupled coefficients, where m is the mesh size and n is the space dimension of lattice points.

Attractors and dimension of dissipative lattice systems

In this paper, by using the argument in [Q.F. Ma, S.H. Wang, C.K. Zhong, Necessary and sufficient conditions for the existence of global attractor for semigroup and application, Indiana Univ. Math. J., 51(6) (2002), 1541–1559.], we prove that the condition given in [S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations 200 (2004) 342–368.] for the existence of a global attractor for the semigroup associated with general lattice systems on a discrete Hilbert space is a sufficient and necessary condition. As an application, we consider the existence of a global attractor for a second-order lattice system in a discrete weighted space containing all bounded sequences. Finally, we show that the global attractor for first-order and partly dissipative lattice systems corresponding to (partly dissipative) reaction–diffusion equations and second-order dissipative lattice systems corresponding to the strongly damped wave equations have finite fractal dimension if the derivative of the nonlinear term is small at the origin.

Attractors for partly dissipative lattice dynamic systems in l2×l2

In this paper, we study the asymptotic behavior of solutions for the partly dissipative lattice dynamical systems in l^2xl^2. We prove the asymptotic compactness of the solutions and then establish...

Cold collisions in dissipative optical lattices

In the past, light-assisted cold collisions between laser cooled atoms have been widely studied in magneto-optical atom traps (MOTs). We describe here theoretical studies of dynamical interactions, specifically cold collisions, between atoms trapped in near-resonant, dissipative optical lattices. The developed quantum-mechanical model is based on Monte Carlo wave-function simulations and combines atomic cooling and collision dynamics in a single framework. It turns out, that the radiative heating mechanism affects the dynamics of atomic cloud in a red-detuned lattice in a way that is not directly expected from the MOT studies. The optical lattice and position dependent light-matter coupling introduces selectivity of collision partners. Atoms, which are most mobile and energetic, are strongly favored to participate in collisions, and are more often ejected from the lattice, than the slow ones in the laser parameter region selected for study. For blue-detuned lattices, we study how optical shielding emerges as a natural part of the lattice and look for ways to optimize the effect. The simulations are computationally very demanding and would obviously benefit from the simplification schemes. We present some steps to this direction by showing how it is possible to calculate collision rates in near-resonant lattices in a fairly simple way.

Resonant Activation in a Nonadiabatically Driven Optical Lattice

We demonstrate the phenomenon of resonant activation in a nonadiabatically driven dissipative optical lattice with broken time symmetry. The resonant activation results in a resonance as a function of the driving frequency in the current of atoms through the periodic potential. We demonstrate that the resonance is produced by the interplay between deterministic driving and fluctuations, and we also show that by changing the frequency of the driving it is possible to control the direction of the diffusion.

Mode-locking of mobile discrete breathers

We study numerically synchronization phenomena of mobile discrete breathers in dissipative nonlinear lattices periodically forced. When varying the driving intensity, the breather velocity generically locks at rational multiples of the driving frequency. In most cases, the locking plateau coincides with the linear stability domain of the resonant mobile breather and desynchronization occurs by the regular appearance of type-I intermittencies. However, some plateaus also show chaotic mobile breathers with locked velocity in the locking region. The addition of a small subharmonic driving tames the locked chaotic solution and enhances the stability of resonant mobile breathers.

Attractors for partly dissipative lattice dynamic systems in [formula omitted

In this paper, we study the asymptotic behavior of solutions for the partly dissipative lattice dynamical systems in l 2 × l 2 . We prove the asymptotic compactness of the solutions and then establish the existence of the global attractor in l 2 × l 2 . The upper semicontinuity of the global attractor is also considered by finite-dimensional approximations of attractors for the lattice systems.

Bloch oscillations in optical dissipative lattices

We show that Bloch oscillations are possible in dissipative optical waveguide lattices with a linearly varying propagation constant. These oscillations occur in spite of the fact that the Bloch wave packet experiences coupling gain and (or) loss. Experimentally, this process can be observed in different settings, such as in laser arrays and lattices of semiconductor optical amplifiers. In addition, we demonstrate that these systems can suppress instabilities arising from preferential mode noise growth.

Rectifying fluctuations in an optical lattice.

We have realized a Brownian motor by using cold atoms in a dissipative optical lattice as a model system. In our experiment the optical potential is spatially symmetric and the time symmetry of the system is broken by applying appropriate zero-mean ac forces. We identify a regime of rectification of forces and a regime of rectification of fluctuations, the latter corresponding to the realization of a Brownian motor.

Dissipative photonic lattice solitons.

We show that discrete dissipative optical lattice solitons are possible in waveguide array configurations that involve periodically patterned semiconductor optical amplifiers and saturable absorbers. The characteristics of these low-power soliton states are investigated, and their propagation constant eigenvalues are mapped on Floquet-Bloch band diagrams. The prospect of observing such low-power dissipative lattice solitons is discussed in detail.