Bifurcation of nonlinear bound states in the periodic Gross-Pitaevskii equation with PT-symmetry

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