Nonlinear dynamics and solitons in optical lattices

Yuri Kivshar

Nonlinear Physics Centre, Research School of Physical Sciences and Engineering,
The Australian National University Canberra ACT 0200, Australia

We describe the nonlinear dynamics and spatially localized states of the Bose-Einstein condensate loaded onto an optical lattice, in the framework of the mean-field approximation. In particular, we predict the existence of spatially localized topological states in the repulsive Bose-Einstein condensate confined by an optical lattice, matter-wave gap vortices, existing in the gaps of the matter-wave bandgap spectrum due to the Bragg scattering. We discuss the structure, stability, and formation dynamics of the gap vortices in the case of two-dimensional optical lattices. We show that such vortices can exist as asymmetric states in ideally symmetric periodic lattices, including rhomboid, rectangular, and triangular vortex solitons on a square lattice.