Ground states of nonlinear Schrodinger systems with saturable nonlinearity for two counterpropagatin
Abstract:
Counterpropagating optical beams in nonlinear media give rise to a host of interesting nonlinear phenomena such as the formation of spatial solitons, spatiotemporal instabilities, self-focusing and self-trapping, etc. Here we study the existence of ground state (the energy minimizer under the L2-normalization condition) in two-dimensional (2D) nonlinear Schr¨odinger (NLS) systems with saturable nonlinearity, which describes paraxial counterpropagating beams in isotropic local media. The nonlinear coefficient of saturable nonlinearity exhibits a threshold which is crucial in determining whether the ground state exists. The threshold can be estimated by the Gagliardo-Nirenberg inequality and the ground state existence can be proved by the energy method, but not the concentration compactness method. Our results also show the essential difference between 2D NLS equations with cubic and saturable nonlinearities.
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