Dissipative optical Bloch waves.

We show that nonlinear optical cavities under periodic modulations can exhibit dissipative Bloch waves. This can be achieved by a proper choice of the incident field in the cavity that must both compensate for losses and match with the modulation period. Dissipative Bloch waves, unlike their conservative counterparts, spontaneously emerge in the cavity and hence correspond to attracting solutions. This makes it possible to experimentally visualize the band structure of the cavity medium. As an illustration of this phenomenon, we performed our analytical investigations on a degenerate optical parametric oscillator with a modulated transverse refractive index.

Field patterns in periodically modulated optical parametric amplifiers and oscillators.

Spatially localized and periodic field patterns in periodically modulated optical parametric amplifiers and oscillators are studied. In the degenerate case (equal signal and idler beams) we elaborate on the systematic method of construction of the stationary localized modes in the amplifiers, and study their properties and stability. We describe a method of constructing periodic solutions in optical parametric oscillators, by adjusting the form of the external driven field to the given form of either signal or pump beams.

Nature of the intrinsic relation between Bloch-band tunneling and modulational instability.

In an example of Bose-Einstein condensates embedded in two-dimensional optical lattices, we show that in nonlinear periodic systems modulational instability and interband tunneling are intrinsically related phenomena. By direct numerical simulations we find that tunneling results in attenuation or enhancement of instability. On the other hand, instability results in asymmetric nonlinear tunneling. The effect strongly depends on the band gap structure and it is especially significant in the case of the resonant tunneling. The symmetry of the coherent structures emerging from the instability reflects the symmetry of both the stable and the unstable states between which the tunneling occurs. Our results provide evidence of the profound effect of the band structure on the superfluid-insulator transition.

Stable and unstable vector dark solitons of coupled nonlinear Schrödinger equations: application to two-component Bose-Einstein condensates.

The dynamics of vector dark solitons in two-component Bose-Einstein condensates is studied within the framework of coupled one-dimensional nonlinear Schrödinger (NLS) equations. We consider the small-amplitude limit in which the coupled NLS equations are reduced to coupled Korteweg-de Vries (KdV) equations. For a specific choice of the parameters the obtained coupled KdV equations are exactly integrable. We find that there exist two branches of (slow and fast) dark solitons corresponding to the two branches of the sound waves. Slow solitons, corresponding to the lower branch of the acoustic wave, appear to be unstable and transform during the evolution into stable fast solitons (corresponding to the upper branch of the dispersion law). Vector dark solitons of arbitrary depths are studied numerically. It is shown that effectively different parabolic traps, to which the two components are subjected, cause an instability of the solitons, leading to a splitting of their components and subsequent decay. A simple phenomenological theory, describing the oscillations of vector dark solitons in a magnetic trap, is proposed.

Adiabatic dynamics of periodic waves in Bose-Einstein condensates with time dependent atomic scattering length.

Evolution of periodic matter waves in one-dimensional Bose-Einstein condensates with time-dependent scattering length is described. It is shown that variation of the effective nonlinearity is a powerful tool for controlled generation of bright and dark solitons starting with periodic waves.