Tunable band-gap structure and gap solitons in the generalized Gross-Pitaevskii equation with a periodic potential.

The tunable band-gap structure is fundamentally important in the dynamics of both linear and nonlinear modes trapped in a lattice because Bloch modes can only exist in the bands of the periodic system and nonlinear modes associating with them are usually confined to the gaps. We reveal that when a momentum operator is introduced into the Gross-Pitaevskii equation (GPE), the bandgap spectra of the periodic system can be shifted upward parabolically by the growth of the constant momentum coefficient. During this process, the band edges become asymmetric, in sharp contrast to the standard GPE with an external periodic potential. Extended complex Bloch modes with asymmetric profiles can be derived by applying a phase transformation to the symmetric profiles. We find that the inherent parity-time symmetry of the complex system is never broken with increasing momentum coefficient. Under repulsive interactions, solitons with different numbers of peaks bifurcating from the band edges are found in finite gaps. We also address the existence of embedded solitons in the generalized two-dimensional GPE. Linear stability analysis corroborated by direct evolution simulations demonstrates that multi-peaked solitons are almost completely stable in their entire existence domains.