2014/05/27 by Clément Gallo, Clement Gallo, Gallo, Clement +3
Engineering · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Cold Atom Physics and Bose-Einstein Condensates #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Photonic Systems #Pattern Formation and Solitons (nlin.PS) #Quantum Mechanics and Non-Hermitian Physics #Terahertz technology and applications #math.AP #math.CA #nlin.PS #nlin.SI
paper · pdf · doi:10.48550/arxiv.1405.7072
20 pages, 4 figures
arxiv created 2014/05/27 · openalex publication_date 2014/05/27 · arxiv updated 2014/05/29 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
For the stationary Gross-Pitaevskii equation with harmonic real and linear imaginary potentials in the space of one dimension, we study the ground state in the limit of large densities (large chemical potentials), where the solution degenerates into a compact Thomas-Fermi approximation. We prove that the Thomas-Fermi approximation can be constructed with an invertible coordinate transformation and an unstable manifold theorem for a planar dynamical system. The Thomas-Fermi approximation can be justified by reducing the existence problem to the Painlevé-II equation, which admits a unique global Hastings-McLeod solution. We illustrate numerically that an iterative approach to solving the existence problem converges but give no analytical proof of this result. Generalizations are discussed for the stationary Gross-Pitaevskii equation with harmonic real and localized imaginary potentials.