An Integrable Model for Density-Modulated Quantum Condensates

An integrable model possessing inhomogeneous ground states is proposed. The model is a higher-order analog of the nonlinear Schr\"odinger equation. The inverse scattering theory with the elliptic-functional background is formulated, and the N-soliton solution is obtained as a particular solution. We provide the exact bosonic and fermionic quasiparticle eigenstates and reveal various kinds of dynamics such as dark soliton billiards, dislocations, gray solitons, and envelope solitons.