Control of the symmetry breaking in double-well potentials by the resonant nonlinearity management.

We introduce a one-dimensional model of Bose-Einstein condensates (BECs), combining the double-well potential, which is a usual setting for the onset of spontaneous-symmetry-breaking (SSB) effects, and time-periodic modulation of the nonlinearity, which may be implemented by means of the Feshbach-resonance-management (FRM) technique. Both cases of the nonlinearity that is repulsive or attractive on the average are considered. In the former case, the main effect produced by the application of the FRM is spontaneous self-trapping of the condensate in either of the two potential wells in parameter regimes where it would remain untrapped in the absence of the management. In the weakly nonlinear regime, the frequency of intrinsic oscillations in the FRM-induced trapped state is very close to half the FRM frequency, suggesting that the effect is accounted for by a parametric resonance. In the case of the attractive nonlinearity, the FRM-induced effect is the opposite, i.e., enforced detrapping of a state which is self-trapped in its unmanaged form. In the latter case, the frequency of oscillations of the untrapped mode is close to a quarter of the driving frequency, suggesting that a higher-order parametric resonance may account for this effect.

Matter-wave dark solitons: stochastic versus analytical results.

The dynamics of dark matter-wave solitons in elongated atomic condensates are discussed at finite temperatures. Simulations with the stochastic Gross-Pitaevskii equation reveal a noticeable, experimentally observable spread in individual soliton trajectories, attributed to inherent fluctuations in both phase and density of the underlying medium. Averaging over a number of such trajectories (as done in experiments) washes out such background fluctuations, revealing a well-defined temperature-dependent temporal growth in the oscillation amplitude. The average soliton dynamics is well captured by the simpler dissipative Gross-Pitaevskii equation, both numerically and via an analytically derived equation for the soliton center based on perturbation theory for dark solitons.

Targeted transfer of solitons in continua and lattices.

We propose a robust mechanism of targeted energy transfer along a line, as well as on a surface, in the form of transport of coherent solitary-wave structures, driven by a moving, spatially localized external ac field ("arm") in a lossy medium. The efficiency and robustness of the mechanism are demonstrated analytically and numerically in terms of the nonlinear Schrödinger (NLS) equation, and broad regions of stable operation are identified in the model's parameter space. Direct simulations show that the driving arm can manipulate solitons equally well in a lattice NLS model. A salient feature, which is revealed by simulations and explained analytically, is a resonant character of the operation of the driving arm in the lattice medium, both integer and fractional resonances being identified. Numerical experiments also demonstrate that the same solitary-wave-transport mechanism works well in two-dimensional lattice media.

One- and two-dimensional solitons in second-harmonic-generating lattices.

In a model of a dynamical lattice with the on-site second-harmonic-generating nonlinearity and harmonic intersite couplings (that may be equal or different for the fundamental and second harmonics), various solitary-wave solutions are considered in one and two dimensions (1D and 2D). Fundamental (single-hump) solitons are identified in either dimension and their stability is examined and compared to previous results as well as to what is known for the model's continuum counterpart. Stability limits in terms of the coupling constants, which depend on the value of the phase-mismatch parameter, are found for solitons of the twisted-mode type in the 1D lattice, and for their counterparts of two different types (one being a discrete vortex) in the 2D lattice. When the twisted-mode soliton is unstable, the instability, which may be either oscillatory or due to imaginary eigenfrequency pairs, transforms the unstable soliton into a stable fundamental one, in both 1D and 2D cases.

Multichannel pulse dynamics in a stabilized Ginzburg-Landau system.

We study the stability and interactions of chirped solitary pulses in a system of nonlinearly coupled cubic Ginzburg-Landau (CGL) equations with a group-velocity mismatch between them, where each CGL equation is stabilized by linearly coupling it to an additional linear dissipative equation. In the context of nonlinear fiber optics, the model describes transmission and collisions of pulses at different wavelengths in a dual-core fiber, in which the active core is furnished with bandwidth-limited gain, while the other, passive (lossy) one is necessary for stabilization of the solitary pulses. Complete and incomplete collisions of pulses in two channels in the cases of anomalous and normal dispersion in the active core are analyzed by means of perturbation theory and direct numerical simulations. It is demonstrated that the model may readily support fully stable pulses whose collisions are quasielastic, provided that the group-velocity difference between the two channels exceeds a critical value. In the case of quasielastic collisions, the temporal shift of pulses, predicted by the analytical approach, is in semiquantitative agreement with direct numerical results in the case of anomalous dispersion (in the opposite case, the perturbation theory does not apply). We also consider a simultaneous collision between pulses in three channels, concluding that this collision remains quasielastic, and the pulses remain completely stable. Thus, the model may be a starting point for the design of a stabilized wavelength-division-multiplexed transmission system.

Ring solitons on vortices.

Interaction of a ring dark or antidark soliton (RDS and RADS, respectively) with a vortex is considered in the defocusing nonlinear Schrödinger equation with cubic (for RDS) or saturable (for RADS) nonlinearities. By means of direct simulations, it is found that the interaction gives rise to either an almost isotropic or a spiral-like pattern. A transition between them occurs at a critical value of the RDS or RADS amplitude, the spiral pattern appearing if the amplitude exceeds the critical value. An initial ring soliton created on top of the vortex splits into a pair of rings moving inward and outward. In the subcritical case, the inbound ring reverses its polarity, bouncing from the vortex core, without conspicuous effect on the core. In the transcritical case, the bounced ring soliton suffers a spiral deformation, while the vortex changes its position and structure and also loses its axial symmetry. Through a variational-type approach to the system's Hamiltonian, we additionally find that the vortex-RDS and vortex-RADS interactions are, respectively, attractive and repulsive. Simulations with the vortex placed eccentrically with respect to the RDS or RADS reveal the generation of strongly localized multispot dark and/or antidark coherent structures. The occurrence of spiral-like patterns in many numerical experiments prompted an attempt to generate a spiral dark soliton, but the latter is found to suffer a core instability that converts it into a rotating dipole emitting waves in the outward direction.

Collisions between spatiotemporal solitons of different dimensionality in a planar waveguide.

A (2+1)-dimensional nonlinear Schrödinger equation including third-order dispersion is a natural model of a waveguide, in which strong temporal dispersion is induced by a grating in order to make the existence of two-dimensional spatiotemporal solitons possible. By means of analytical and numerical methods, we demonstrate that this model may support, simultaneously, stable dark quasi-one-dimensional (stripe) solitons and two-dimensional elevation solitons ("antidark solitons") in the form of weakly localized "lumps." The spatial position of lumps can be controlled by passing stripe dark solitons through them in an arbitrary direction. To substantiate this mechanism, we analytically calculate a position shift generated by a headon collision between the stripe and lump. The obtained results are in good agreement with direct numerical simulations.

Dissipative solitons under the action of the third-order dispersion.

We study the evolution of a solitary pulse in the cubic complex Ginzburg-Landau equation, including the third-order dispersion (TOD) as a small perturbation. We develop analytical approximations, which yield a TOD-induced velocity c of the pulse as a function of the ratio D of the second-order dispersion and filtering coefficients. The analytical predictions show agreement with the direct numerical simulations for two distinct intervals of D. A new feature of the pulse motion, which is a precursor of the transition to blowup, is presented: The pulse suddenly acquires a large acceleration in the reverse direction at D>D(cr) approximately -1.5 and without the reversal at D