arXiv subjects
Publications and source records attributed to Clement Gallo.
We prove existence and uniqueness of a positive solution to a system of two coupled Gross-Pitaevskii equations. We give a full asymptotic expansion of this solution into powers of the semi classical parameter $\varepsilon$ in the Thomas--Fermi limit $\varepsilon\to 0$.
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.
From the asymptotic expansion of the ground state of the Gross-Pitaevskii equation in the Thomas--Fermi limit given by Gallo and Pelinovsky in a previous work, we infer an asymptotic expansion of the kinetic, potential and total energy of the ground state. In particular, we give a rigorous proof of the expansion of the kinetic energy calculated by Dalfovo, Pitaevskii and Stringari in the case where the space dimension is 3. Moreover, we calculate one more term in this expansion, and we generalize the result to space dimensions 1 and 2.