Discrete Solitons and Discrete Vortices in (PT-symmetric) Nonlinear Schrodinger Lattices
Tuesday, October 4, 2016 - 11:00am - 12:00pm
Haitao Xu (University of Minnesota, Twin Cities)
Nonlinear lattice dynamical systems in one, two, and three dimensions describe a wide class of interesting models and find applications of significant importance in various fields. In particular, coherent structures like discrete solitons and discrete vortices in nonlinear Schrodinger lattices have emerged in the studies of i.e. photorefractive crystals in nonlinear optics and droplets of optical lattices in Bose-Einstein condensates. In this talk, we investigate the existence and stability of such localized modes in the discrete nonlinear Schrodinger (dNLS) equations. We start from the well-understood anti-continuum (AC) limit and select elementary sets of lattice nodes that can support desired modes in the limit. By employing Lyapunov–Schmidt reductions, the continuation of the discrete solitons and dicrete vortices for small coupling between lattice nodes can be obtained. By embedding the small elementary sets back to the lattice, the persistence and stability results on local cells can be extended to array of lattice cells. We consider the Hamiltonian case as well as non-Hamiltonian case where gains and losses are imposed on lattice sites in a balanced fashion, satisfying the so-called Parity-Time-symmetry.