Conserved Gross–Pitaevskii equations with a parabolic potential. (26th September 2022) - Liu, Shi-min; Zhang, Da-jun
Abstract: An integrable Gross–Pitaevskii equation with a parabolic potential is presented where particle density ∣ u ∣ 2 is conserved. We also present an integrable vector Gross–Pitaevskii system with a parabolic potential, where the total particle density ∑ j = 1 n ∣ u j ∣ 2 is conserved. These equations are related to nonisospectral scalar and vector nonlinear Schrödinger equations. Infinitely many conservation laws are obtained. Gauge transformations between the standard isospectral nonlinear Schrödinger equations and the conserved Gross–Pitaevskii equations, both scalar and vector cases are derived. Solutions and dynamics are analyzed and illustrated. Some solutions exhibit features of localized-like waves.