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Trends in Computational and Applied Mathematics

Mathematical analysis of a Mateo-Delgado model for cigar-shaped Bose-Einstein condensates

R. Cipolatti, Y de Macedo Lira, G. Saisse

Published Jul 31, 2026• 0 citations
Publisher

Abstract

In this paper we present a mathematical analysis of one-dimensional effective model proposed by A.M.~Mateo and V.~Delgado concerning Bose-Einstein condensates in the presence of harmonic confinement. Among the properties presented, we mention: existence, ``uniqueness'', orbital stability, symmetry and gaussian asymptotic decay of ground-state solutions in the repulsive case. We also report formul\ae\ for the minimal energy $E_{\mn}$ and the associate chemical potential $\mu$ as functions of $\lambda$, which is a parameter related to $N$ (the number of atoms) and/or $a$ (the s-wave scattering length). By considering Taylor's development of the non-quadratic therm of the energy and using appropriate gaussian functions as approximations for the ground state, we present some numerical experiments to illustrate our results

DOI
10.5540/tcam.2026.027.e01918

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In this paper we present a mathematical analysis of one-dimensional effective model proposed by A.M.~Mateo and V.~Delgado concerning Bose-Einstein condensates in the presence of harmonic confinement. Among the properties presented, we mention: existence, ``uniqueness'', orbital stability, symmetry and gaussian asymptotic decay of ground-state solutions in the repulsive case. We also report formul\ae\ for the minimal energy E\mnE_{\mn}E\mn​ and the associate chemical potential μ\muμ as functions of λ\lambdaλ, which is a parameter related to NNN (the number of atoms) and/or aaa (the s-wave scattering length). By considering Taylor's development of the non-quadratic therm of the energy and using appropriate gaussian functions as approximations for the ground state, we present some numerical experiments to illustrate our results
In this paper we present mathematical analysis of one-dimensional effective models proposed in [\cite{MunozDelgado}] concerning Bose-Einstein condensates in the presence of harmonic confinement. Among the demonstrated properties, we can mention: existence, uniqueness, orbital stability, symmetry and gaussian asymptotic decay of ground-state solutions in the repulsive case. We also report formulæ for the minimal energy $E_{\mn}$ and the associate chemical potential μμμ as functions of a parameter λλλ, which is related to NNN (the number of atoms) and/or aaa (the s-wave scattering length). By considering Taylor's development of the non-quadratic therm of the energy and using appropriate gaussian functions as approximations for the ground state, we present some numerical experiments to illustrate our results.
The Gross-Pitaevskii equation for a Bose-Einstein condensate confined in an elongated cigar-shaped trap is reduced to an effective system of nonlinear equations depending on only one space coordinate along the trap axis. The radial distribution of the condensate density and its radial velocity are approximated by Gaussian functions with real and imaginary exponents, respectively, with parameters depending on the axial coordinate and time. The effective one-dimensional system is applied to a description of the ground state of the condensate, to dark and bright solitons, to the sound and radial compression waves propagating in a dense condensate, and to weakly nonlinear waves in repulsive condensate. In the low-density limit our results reproduce the known formulas. In the high-density case our description of solitons goes beyond the standard approach based on the nonlinear Schr\"odinger equation. The dispersion relations for the sound and radial compression waves are obtained in a wide region of values of the condensate density. The Korteweg--de Vries equation for weakly nonlinear waves is derived and the existence of bright solitons on a constant background is predicted for a dense enough condensate with a repulsive interaction between the atoms.
In this paper we study the effects of the transverse confinement strength on the dark solitons dynamics in quasi-one-dimensional Bose–Einstein condensates (BECs). The theoretical model is based on the Muoz-Mateo & Delgado (MMD) equation that describes cigar-shaped BECs with repulsive interatomic interactions. Since the MMD equation presents a nonpolynomial form, the soliton-sound recombination may not display the same pattern presented in the cubic model. We perform numerical simulations for harmonic and anharmonic (Gaussian) external potentials to compare both cases. All results via the MMD equation are in perfect agreement with the 3D simulations. Also, we noticed some differences between MMD and cubic models and changes in the soliton lifetime due to the values of the transverse confinement.
We consider the stability and dynamics of multiple dark solitons in cigar-shaped Bose-Einstein condensates. Our study is motivated by the fact that multiple matter-wave dark solitons may naturally form in such settings as per our recent work [Phys. Rev. Lett. 101, 130401 (2008)]. First, we study the dark soliton interactions and show that the dynamics of well-separated solitons (i.e., ones that undergo a collision with relatively low velocities) can be analyzed by means of particle-like equations of motion. The latter take into regard the repulsion between solitons (via an effective repulsive potential) and the confinement and dimensionality of the system (via an effective parabolic trap for each soliton). Next, based on the fact that stationary, well-separated dark multisoliton states emerge as a nonlinear continuation of the appropriate excited eigenstates of the quantum harmonic oscillator, we use a Bogoliubov-de Gennes analysis to systematically study the stability of such structures. We find that for a sufficiently large number of atoms, multiple soliton states are dynamically stable, while for a small number of atoms, we predict a dynamical instability emerging from resonance effects between the eigenfrequencies of the soliton modes and the intrinsic excitation frequencies of the condensate. Finally, we present experimental realizations of multisoliton states including a three-soliton state consisting of two solitons oscillating around a stationary one and compare the relevant results to the predictions of the theoretical mean-field model.
In recent years, the study of cigar shaped Bose?Einstein condensate (BEC) under a variety of external confinements has attracted a great deal of attention, from both theoretical and experimental researchers. We report a unified model for obtaining explicit solutions under various kinds of physically relevant space- and time-modulated external traps and nonlinearities for the cigar shaped BEC. Our novel mechanism paves the way to investigate the system for a family of potential functions unified as a physical parameter of the system. We apply and illustrate our results for a number of exactly solvable quantum mechanical potentials; harmonic, double-well, P?sch?Teller, Morse, Toda lattice and power-law as some of the applications of our model. Expressions of the condensate density are provided for these potentials. In addition, as an application of our model, we have illustrated the condensate dynamics for harmonic, double-well and P?sch?Teller potentials. In the presence of loss/gain for a P?sch?Teller potential, the condensate density shows an interesting collapse and revival.