Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0907.1248 · Journal of Physics A Mathematical and Theoretical

Generalized nonpolynomial Schrödinger equations for matter waves under anisotropic transverse confinement

Luca Salasnich

Published Jul 30, 2009• 30 citations
arXivHTMLPDF

Abstract

Starting from the three-dimensional Gross–Pitaevskii equation we derive a 1D generalized nonpolynomial Schrödinger equation, which describes the dynamics of Bose–Einstein condensates under the action of a generic potential in the longitudinal axial direction and of an anisotropic harmonic potential in the transverse radial direction. This equation reduces to the familiar 1D nonpolynomial Schrödinger equation (Salasnich, Parola and Reatto 2002 Phys. Rev. A 65 , 043614) in the case of isotropic transverse harmonic confinement. In addition, we show that if the longitudinal potential models a periodic optical lattice the 3D GPE can be mapped into a 1D generalized discrete nonpolynomial Schrödinger equation.

DOI
10.1088/1751-8113/42/33/335205

Related papers

Quasi-one-dimensional approximation for Bose–Einstein condensates transversely trapped by a funnel potential

Mateus C. P. dos Santos, Boris A. Malomed, Wesley Bueno Cardoso

Abstract Starting from the standard three-dimensional (3D) Gross–Pitaevskii equation (GPE) and using a variational approximation, we derive an effective one-dimensional nonpolynomial Schrödinger equation (1D-NPSE) governing the axial dynamics of atomic Bose–Einstein condensates (BECs) under the action of a singular but physically relevant funnel-shaped transverse trap, i.e. an attractive 2D potential ∼−1/ r (where r is the radial coordinate in the transverse plane), in combination with the repulsive self-interaction. Wave functions of the trapped BEC are regular, in spite of the potential’s singularity. The model applies to a condensate of particles (small molecules) carrying a permanent electric dipole moment in the field of a uniformly charged axial thread, as well as to a quantum gas of magnetic atoms pulled by an axial electric current. By means of numerical simulations, we verify that the effective 1D-NPSE provides accurate static and dynamical results, in comparison to the full 3D GPE, for both repulsive and attractive signs of the intrinsic nonlinearity.

arXiv:1910.07001•Journal of Physics B Atomic Molecular and Optical Physics•Oct 21, 2019•10 citations

Generalized one-dimensional nonpolynomial Schrödinger equation for Bose-Einstein condensates with generic transverse confinement

Andréia M. Basso, Wesley Bueno Cardoso

This work presents a dimensional reduction of Bose-Einstein condensates confined by generalized transverse potentials, parametrized by an exponent n . Starting from the three-dimensional Gross-Pitaevskii equation, we employ a variational ansatz to derive an effective one-dimensional nonpolynomial Schrödinger equation, which self-consistently determines the transverse width dynamics. The model generalizes existing formalisms for harmonic and funnel-shaped transverse confinement potentials. We validate the approach through comprehensive numerical tests, demonstrating excellent agreement with full 3D simulations for ground-state properties across various interaction regimes. Finally, real-time simulations of matter-wave scattering at potential barriers verify the model’s dynamical robustness, successfully replicating the spatiotemporal evolution and energy-dependent transmission characteristics observed in full 3D calculations.

arXiv:2511.14973•Physics Letters A•Sep 16, 2026•0 citations

1 3D BEC Bright Solitons under Transverse Confinement: Analytical Results with the Nonpolynomial Schrödinger Equation

Luca Salasnich

The Bose-Einstein condensate (BEC) of a dilute gas of bosons is well described by the three-dimensional Gross-Pitaevskii equation (3D GPE), that is a nonlinear Schrödinger equation. By imposing a transverse confinement the BEC can travel only in the cylindrical axial direction. We show that in this case the BEC with attractive interaction admits a 3D bright soliton solution which generalizes the text-book one, that is valid in the one-dimensional limit (1D GPE). Contrary to the 1D case, the 3D bright soliton exists only below a critical number of Bosons that depends on the extent of confinement. Finally, we find that the 3D bright soliton collapses if its density excedes a critical value. Our results are obtained by using a nonpolynomial Schrödinger equation (NPSE), an effective one-dimensional equation derived from the 3D GPE. In a dilute gas of N Bosons at zero temperature practically all particles are in the same single-particle state of the density matrix ρ(r,r ′;t). 1), 2) This macroscopically occupied single-particle state is called Bose-Einstein condensate (BEC) and it is well described by a complex classical field ψ(r,t) (order parameter or macroscopic

•Jan 1, 2002•0 citations

3D BEC Bright Solitons under Transverse Confinement

Luca Salasnich

The Bose-Einstein condensate (BEC) of a dilute gas of bosons is well described by the three-dimensional Gross-Pitaevskii equation (3D GPE), that is a nonlinear Schrödinger equation. By imposing a transverse confinement the BEC can travel only in the cylindrical axial direction. We show that in this case the BEC with attractive interaction admits a 3D bright soliton solution which generalizes the text-book one, that is valid in the one-dimensional limit (1D GPE). Contrary to the 1D case, the 3D bright soliton exists only below a critical number of Bosons that depends on the extent of confinement. Finally, we find that the 3D bright soliton collapses if its density excedes a critical value. Our results are obtained by using a nonpolynomial Schrödinger equation (NPSE), an effective one-dimensional equation derived from the 3D GPE.

arXiv:nlin/0208018•Progress of Theoretical Physics Supplement•Jan 1, 2003•1 citation

Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates

Luca Salasnich, Alberto Parola, L. Reatto

Starting from the three-dimensional (3D) Gross-Pitaevskii equation and using a variational approach, we derive an effective 1D wave equation that describes the axial dynamics of a Bose condensate confined in an external potential with cylindrical symmetry. The trapping potential is harmonic in the transverse direction and generic in the axial one. Our equation, that is a time-dependent nonpolynomial nonlinear Schr\"odinger equation (1D NPSE), can be used to model cigar-shaped condensates, whose dynamics is essentially 1D. We show that 1D NPSE gives much more accurate results than all other effective equations recently proposed. By using 1D NPSE we find analytical solutions for bright and dark solitons, which generalize the ones known in the literature. We deduce also an effective 2D nonpolynomial Schr\"odinger equation (2D NPSE) that models disk-shaped Bose condensates confined in an external trap that is harmonic along the axial direction and generic in the transverse direction. In the limiting cases of weak and strong interaction, our approach gives rise to Schr\"odinger-like equations with different polynomial nonlinearities.

arXiv:cond-mat/0201395•Physical Review A•Apr 2, 2002•673 citations

Gap solitons in elongated geometries: The one-dimensional Gross-Pitaevskii equation and beyond

Antonio Muñoz Mateo, V. Delgado, Boris A. Malomed

We report results of a systematic analysis of matter-wave gap solitons (GSs) in three-dimensional self-repulsive Bose-Einstein condensates (BECs) loaded into a combination of a cigar-shaped trap and axial optical-lattice (OL) potential. Basic cases of the strong, intermediate, and weak radial (transverse) confinement are considered, as well as settings with shallow and deep OL potentials. Only in the case of the shallow lattice combined with tight radial confinement, which actually has little relevance to realistic experimental conditions, does the usual one-dimensional (1D) cubic Gross-Pitaevskii equation (GPE) furnish a sufficiently accurate description of GSs. However, the effective 1D equation with the nonpolynomial nonlinearity, derived in Ref. [Phys. Rev. A 77, 013617 (2008)], provides for quite an accurate approximation for the GSs in all cases, including the situation with weak transverse confinement, when the soliton's shape includes a considerable contribution from higher-order transverse modes, in addition to the usual ground-state wave function of the respective harmonic oscillator. Both fundamental GSs and their multipeak bound states are considered. The stability is analyzed by means of systematic simulations. It is concluded that almost all the fundamental GSs are stable, while their bound states may be stable if the underlying OL potential is deep enough.

arXiv:1104.0962•Physical Review A•May 9, 2011•18 citations
Starting from the three-dimensional Gross–Pitaevskii equation we derive a 1D generalized nonpolynomial Schrödinger equation, which describes the dynamics of Bose–Einstein condensates under the action of a generic potential in the longitudinal axial direction and of an anisotropic harmonic potential in the transverse radial direction. This equation reduces to the familiar 1D nonpolynomial Schrödinger equation (Salasnich, Parola and Reatto 2002 Phys. Rev. A 65 , 043614) in the case of isotropic transverse harmonic confinement. In addition, we show that if the longitudinal potential models a periodic optical lattice the 3D GPE can be mapped into a 1D generalized discrete nonpolynomial Schrödinger equation.
Abstract Starting from the standard three-dimensional (3D) Gross–Pitaevskii equation (GPE) and using a variational approximation, we derive an effective one-dimensional nonpolynomial Schrödinger equation (1D-NPSE) governing the axial dynamics of atomic Bose–Einstein condensates (BECs) under the action of a singular but physically relevant funnel-shaped transverse trap, i.e. an attractive 2D potential ∼−1/ r (where r is the radial coordinate in the transverse plane), in combination with the repulsive self-interaction. Wave functions of the trapped BEC are regular, in spite of the potential’s singularity. The model applies to a condensate of particles (small molecules) carrying a permanent electric dipole moment in the field of a uniformly charged axial thread, as well as to a quantum gas of magnetic atoms pulled by an axial electric current. By means of numerical simulations, we verify that the effective 1D-NPSE provides accurate static and dynamical results, in comparison to the full 3D GPE, for both repulsive and attractive signs of the intrinsic nonlinearity.
This work presents a dimensional reduction of Bose-Einstein condensates confined by generalized transverse potentials, parametrized by an exponent n . Starting from the three-dimensional Gross-Pitaevskii equation, we employ a variational ansatz to derive an effective one-dimensional nonpolynomial Schrödinger equation, which self-consistently determines the transverse width dynamics. The model generalizes existing formalisms for harmonic and funnel-shaped transverse confinement potentials. We validate the approach through comprehensive numerical tests, demonstrating excellent agreement with full 3D simulations for ground-state properties across various interaction regimes. Finally, real-time simulations of matter-wave scattering at potential barriers verify the model’s dynamical robustness, successfully replicating the spatiotemporal evolution and energy-dependent transmission characteristics observed in full 3D calculations.
The Bose-Einstein condensate (BEC) of a dilute gas of bosons is well described by the three-dimensional Gross-Pitaevskii equation (3D GPE), that is a nonlinear Schrödinger equation. By imposing a transverse confinement the BEC can travel only in the cylindrical axial direction. We show that in this case the BEC with attractive interaction admits a 3D bright soliton solution which generalizes the text-book one, that is valid in the one-dimensional limit (1D GPE). Contrary to the 1D case, the 3D bright soliton exists only below a critical number of Bosons that depends on the extent of confinement. Finally, we find that the 3D bright soliton collapses if its density excedes a critical value. Our results are obtained by using a nonpolynomial Schrödinger equation (NPSE), an effective one-dimensional equation derived from the 3D GPE. In a dilute gas of N Bosons at zero temperature practically all particles are in the same single-particle state of the density matrix ρ(r,r ′;t). 1), 2) This macroscopically occupied single-particle state is called Bose-Einstein condensate (BEC) and it is well described by a complex classical field ψ(r,t) (order parameter or macroscopic
The Bose-Einstein condensate (BEC) of a dilute gas of bosons is well described by the three-dimensional Gross-Pitaevskii equation (3D GPE), that is a nonlinear Schrödinger equation. By imposing a transverse confinement the BEC can travel only in the cylindrical axial direction. We show that in this case the BEC with attractive interaction admits a 3D bright soliton solution which generalizes the text-book one, that is valid in the one-dimensional limit (1D GPE). Contrary to the 1D case, the 3D bright soliton exists only below a critical number of Bosons that depends on the extent of confinement. Finally, we find that the 3D bright soliton collapses if its density excedes a critical value. Our results are obtained by using a nonpolynomial Schrödinger equation (NPSE), an effective one-dimensional equation derived from the 3D GPE.
Starting from the three-dimensional (3D) Gross-Pitaevskii equation and using a variational approach, we derive an effective 1D wave equation that describes the axial dynamics of a Bose condensate confined in an external potential with cylindrical symmetry. The trapping potential is harmonic in the transverse direction and generic in the axial one. Our equation, that is a time-dependent nonpolynomial nonlinear Schr\"odinger equation (1D NPSE), can be used to model cigar-shaped condensates, whose dynamics is essentially 1D. We show that 1D NPSE gives much more accurate results than all other effective equations recently proposed. By using 1D NPSE we find analytical solutions for bright and dark solitons, which generalize the ones known in the literature. We deduce also an effective 2D nonpolynomial Schr\"odinger equation (2D NPSE) that models disk-shaped Bose condensates confined in an external trap that is harmonic along the axial direction and generic in the transverse direction. In the limiting cases of weak and strong interaction, our approach gives rise to Schr\"odinger-like equations with different polynomial nonlinearities.
We report results of a systematic analysis of matter-wave gap solitons (GSs) in three-dimensional self-repulsive Bose-Einstein condensates (BECs) loaded into a combination of a cigar-shaped trap and axial optical-lattice (OL) potential. Basic cases of the strong, intermediate, and weak radial (transverse) confinement are considered, as well as settings with shallow and deep OL potentials. Only in the case of the shallow lattice combined with tight radial confinement, which actually has little relevance to realistic experimental conditions, does the usual one-dimensional (1D) cubic Gross-Pitaevskii equation (GPE) furnish a sufficiently accurate description of GSs. However, the effective 1D equation with the nonpolynomial nonlinearity, derived in Ref. [Phys. Rev. A 77, 013617 (2008)], provides for quite an accurate approximation for the GSs in all cases, including the situation with weak transverse confinement, when the soliton's shape includes a considerable contribution from higher-order transverse modes, in addition to the usual ground-state wave function of the respective harmonic oscillator. Both fundamental GSs and their multipeak bound states are considered. The stability is analyzed by means of systematic simulations. It is concluded that almost all the fundamental GSs are stable, while their bound states may be stable if the underlying OL potential is deep enough.