Rogue waves and nonzero background solutions for the Gross-Pitaevskii equation with a parabolic potential

In this paper an integrable Gross–Pitaevskii equation with a parabolic potential and a gain term is investigated. Its solutions with a nonzero background are derived. These solutions are constructed by using biliearization reduction approach and connections between the nonlinear Schrödinger equation and the Gross–Pitaevskii equation. The solutions are presented in double-Wronskian form and are classified in terms of canonical forms of a certain matrix. Various breathers and rogue waves are analyzed and illustrated.