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

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

Bifurcation of nonlinear bound states in the periodic Gross-Pitaevskii equation with ��-symmetry Journal Articles