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T. Dohnal and D. Pelinovsky

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

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**Abstract:**

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.

**Keywords**:

Gross-Pitaevskii equation, parity-time reversal symmetry, periodic potentials,
effective amplitude equations, Lyapunov-Schmidt reductions.