Integrable model for density-modulated quantum condensates: solitons passing through a soliton lattice

An integrable model possessing inhomogeneous ground states is proposed as an effective model of non-uniform quantum condensates such as supersolids and Fulde--Ferrell--Larkin--Ovchinnikov superfluids. The model is a higher-order analog of the nonlinear Schr\"odinger equation. We derive an $n$-soliton solution via the inverse scattering theory with elliptic-functional background, and reveal various kinds of soliton dynamics such as dark soliton billiards, dislocations, gray solitons, and envelope solitons. We also provide the exact bosonic and fermionic quasiparticle eigenstates and clarify their tunneling phenomena. The solutions are expressed by a determinant of theta functions.