Conformal duality of Bose-Einstein condensates with two- and three-body interactions

Solitary waves and droplets are intriguing phenomena that fundamentally rely on the nonlinear atom-atom interactions of Bose-Einstein condensates (BECs). Remarkably, in the quasi-one dimensional case analytical solutions exist for the cubic nonlinear Schrödinger equation used to describe BECs with two-body interactions [1]. However, new solution types emerge when we consider higher-order interactions such as three-body interactions. Here, we show that the cubic-quintic nonlinear Schrödinger equation exhibits a unique correspondence to its lower-order counterpart in terms of a conformal duality. By means of this duality we relate the densities and the velocity fields of Bose-condensed systems with and without three-body interactions allowing us to determine the properties of a BEC by means of its conformal partner with corresponding higher- or lower interaction order. We explain the origin, present some applications and show the generalization to arbitrary high-order scattering processes within the mean field regime including the case of the non-interacting linear Schrödinger equation. Finally, we discuss the applicability of the conformal duality to higher-dimensional systems.

[1] L.D. Carr, C.W. Clark, W.P. Reinhardt, Phys. Rev. A 62, 063610 & 063611 (2000)