Nonlinear Schrödinger equations for Bose-Einstein condensates

Abstract

The Gross-Pitaevskii equation, or more generally the nonlinear Schrödinger equation, models the Bose-Einstein condensates in a macroscopic gaseous superfluid wave-matter state in ultra-cold temperature. We provide analytical study of the NLS with L 2 initial data in order to understand propagation of the defocusing and focusing waves for the BEC mechanism in the presence of electromagnetic fields. Numerical simulations are performed for the two-dimensional GPE with anisotropic quadratic potentials.

Universal Upper Bound on The Blowup Rate of Nonlinear Schrödinger Equation with Rotation

“…Our approach allows to technically deal with the commuting issue between x, ∇ and U (t − s), and provides a treatment for general sublinear A and subquadratic V assumed in Proposition 2.1. Special cases of A and V for the RNLS were studied in the literature, see e.g., [3,9,10,14,17].…”

“…For λ ∈ R and 1 ≤ p < 1 + 4 n−2 , the local well-posedness results of equation (1) were obtained in e.g. [3,6,17], see also [10,11,14,27] for the treatment in a general magnetic setting. In the focusing case λ > 0 and p ≥ 1 + 4 n , there exist solutions that blowup in finite time [7,8,23,25,26].…”

Threshold for Blowup and Stability for Nonlinear Schrödinger Equation with Rotation

“…The local wellposedness for V e (x) = − j γ2 j x 2 j follow from a Strichartz estimate in Lemma 2.4, see Proposition 2.1. In the mass-subcritical case p < 1 + 4/n with any data in L 2 and in the mass-critical case p = 1 + 4/n with small data in L 2 , the global in time solution exists and is unique [20,50]…”

“…has been considered in e.g. [2,13,27] when n = 2, 3, and [16,20,39,50] for general magnetic NLS (mNLS), to list a few. RNLS (1) can be written in the magnetic form (5), thus the wellposedness follows from that of mNLS if p ∈ (1, 1 + 4/(n − 2)) in view of Proposition 2.1.…”

Two-breather solutions for the class I infinitely extended nonlinear Schrödinger equation and their special cases

“…The nonlinear Schrödinger equation [1,2] (NLSE) has various applications in describing ocean waves [3,4,5], pulses in optical fibres [6,7,8], Bose-Einstein condensates [9,10,11,12], waves in the atmosphere [13], plasma [14] and many other physical systems [15,16,17,18,19]. Various extensions of the NLSE have been considered [20,21,22] that increase the accuracy of the description of nonlinear wave phenomena in these systems by incorporating higher-order effects [23,24,25,26].…”