Abstract
AbstractThe stationary GrossāPitaevskii equation in one dimension is considered with a complex periodic potential satisfying the conditions of the š«šÆ (parity-time reversal) symmetry. Under rather general assumptions on the potentials, we prove bifurcations of š«šÆ-symmetric nonlinear bound states from the end points of a real interval in the spectrum of the non-selfadjoint linear Schrƶdinger operator with a complex š«šÆ-symmetric periodic potential. The nonlinear bound states are approximated by the effective amplitude equation, which bears the form of the cubic nonlinear Schrƶdinger equation. In addition, we provide sufficient conditions for the appearance of complex spectral bands when the complex š«šÆ-symmetric potential has an asymptotically small imaginary part.
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