Mobility edges and fractal states in quasiperiodic Gross-Pitaevskii chains

We explore the properties of a Gross-Pitaevskii chain subject to an incommensurate periodic potential, i.e., a nonlinear Aubry-Andr\'e model. We show that the condensate crucially impacts the properties of the elementary excitations. In contrast to the conventional linear Aubry-Andr\'e model, the boundary between localized and extended states (mobility edge) exhibits nontrivial branching. For instance, in the high-density regime, ``tongues'' of extended phases at intermediate energies penetrate the domain of localized states. In the low-density case, the situation is opposite, and tongues of localized phases emerge. Moreover, intermediate critical (fractal) states are observed. The low-energy phonon part of the spectrum is robust against the incommensurate potential. Our study shows that accounting for interactions, already at the classical level, leads to highly nontrivial behavior of the elementary excitation spectrum.