Solitary waves in nematic liquid crystals

We study soliton solutions of a two-dimensional nonlocal NLS equation of Hartree-type with a Bessel potential kernel. The equation models laser propagation in nematic liquid crystals. Motivated by the experimental observation of spatially localized beams, see Conti et al. (2003), we show existence, stability, regularity, and radial symmetry of energy minimizing soliton solutions in R2. We also give theoretical lower bounds for the L2-norm (power) of these solitons, and show that small L2 -norm initial conditions lead to decaying solutions. We also present numerical computations of radial soliton solutions. These solutions exhibit the properties expected by the infinite plane theory, although we also see some finite (computational) domain effects, especially solutions with arbitrarily small power.