For a number of nonlocal nonlinear equations such as nonlocal, nonlinear Schrodinger equation (NLSE), nonlocal Ablowitz-Ladik (AL), nonlocal, saturable discrete NLSE (DNLSE), coupled nonlocal NLSE, coupled nonlocal AL and coupled nonlocal, saturable DNLSE, we obtain periodic solutions in terms of Jacobi elliptic functions as well as the corresponding hyperbolic soliton solutions. Remarkably, in all the six cases, we find that unlike the corresponding local cases, all the nonlocal models simultaneously admit both the bright and the dark soliton solutions. Further, in all the six cases, not only \(\rm{Dn}(x,m)\) and \(\rm{Cn}(x,m)\) but even their linear superposition is shown to be an exact solution. Finally, we show that the coupled nonlocal NLSE not only admits solutions in terms of Lame polynomials of order 1, but it also admits solutions in terms of Lame polynomials of order 2, even though they are not the solutions of the uncoupled nonlocal problem. We also remark on the possible integrability in certain cases.

http://arxiv.org/abs/1405.5267
Pattern Formation and Solitons (nlin.PS)

We investigate the parity- and time-reversal (PT)-symmetry breaking in lattice models in the presence of long-ranged, non-hermitian, PT-symmetric potentials that remain finite or become divergent in the continuum limit. By scaling analysis of the fragile PT threshold for an open finite lattice, we show that continuum loss-gain potentials \(V_a(x)\sim i|x|^a{\rm sign}(x)\) have a positive PT-breaking threshold for \(\alpha>−2\), and a zero threshold for α≤−2. When α<0 localized states with complex (conjugate) energies in the continuum energy-band occur at higher loss-gain strengths. We investigate the signatures of PT-symmetry breaking in coupled waveguides, and show that the emergence of localized states dramatically shortens the relevant time-scale in the PT-symmetry broken region.

http://arxiv.org/abs/1403.4204
Quantum Physics (quant-ph); Optics (physics.optics)