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

Journal of the Physical Society of Japan

Solitons in a One-Dimensional Bose System with the Repulsive Delta-Function Interaction

M. Ishikawa, Hajime Takayama

Published Oct 15, 1980• 81 citations
Publisher

Abstract

Lieb's type II excitation, derived exactly by means of a Bethe ansatz, in a one-dimensional Bose system with the repulsive delta-function interaction is identified with the soliton mode, which is a solution of the Pitaevskii-Gross (or nonlinear Schrödinger) equation for the corresponding Bose field.
DOI
10.1143/jpsj.49.1242

Related papers

The dynamics of the one-dimensional delta-function Bose gas

Craig A. Tracy, Harold Widom

We give a method to solve the time-dependent Schrödinger equation for a system of one-dimensional bosons interacting via a repulsive delta function potential. The method uses the ideas of Bethe Ansatz but does not use the spectral theory of the associated Hamiltonian.

arXiv:0808.2491•Journal of Physics A Mathematical and Theoretical•Oct 21, 2008•28 citations

Bose Gas in One Dimension: Lieb–Liniger Model

Hans-Peter Eckle

Abstract The coordinate Bethe ansatz can be extended to a model, the Lieb–Liniger model, of a one-dimensional gas of Bosons interacting with repulsive δ-function potentials. It has attracted attention due to its relevance for experimental developments in the fields of ultracold gases and optical lattices. This chapter provides an exposition of the related classical nonlinear Schrödinger equation, followed by its generalization to the quantum model. It explores a limiting case, the Tonks-Girardeau gas. The δ-function potentials supply a kind of boundary condition on the wave functions allowing us to analyze the eigenfunctions of the Bethe ansatz, which are examined on the infinite line and for periodic boundary conditions. The latter leads to the Bethe ansatz equations. The solution of these equations is achieved in the thermodynamic limit for the ground state and for low-lying excited states.

•Jul 25, 2019•0 citations

Solitons in Bose-Einstein Condensates

David Pascual Solis

Investigation of 1D-(bright) solitons in Bose-Einstein Condensates beyond mean-field theory (Gross-Pitaevskii equation) using quantum field theoretic approaches. * Bethe ansatz method provides exact solution for zero-range pseudo interaction potential. * Discussion of a more general model in the dilute regime for taking into account more realistic atomic interactions. Among others: Computeralgebraic implementation of a commutator algebra.

Science•Jan 7, 2000•1 citation

Beyond Gross-Pitaevskii equation for 1D gas: quasiparticles and solitons

Jakub Kopyciński, Maciej Łebek, Maciej Marciniak +3 more

Describing properties of a strongly interacting quantum many-body system poses a serious challenge both for theory and experiment. In this work, we study excitations of one-dimensional repulsive Bose gas for arbitrary interaction strength using a hydrodynamic approach. We use linearization to study particle (type-I) excitations and numerical minimization to study hole (type-II) excitations. We observe a good agreement between our approach and exact solutions of the Lieb-Liniger model for the particle modes and discrepancies for the hole modes. Therefore, the hydrodynamical equations find to be useful for long-wave structures like phonons and of a limited range of applicability for short-wave ones like narrow solitons. We discuss potential further applications of the method.

arXiv:2106.15289•SciPost Physics•Jan 14, 2022•7 citations

Single-shot simulations of dynamics of quantum dark solitons

Andrzej Syrwid, Mirosław Brewczyk, Mariusz Gajda +1 more

Eigenstates of Bose particles with repulsive contact interactions in one-dimensional space with periodic boundary conditions can be found with the help of the Bethe ansatz. The type II excitation spectrum identified by E. H. Lieb, reproduces the dispersion relation of dark solitons in the mean-field approach. The corresponding eigenstates possess translational symmetry which can be broken in measurements of positions of particles. We analyze emergence of single and double solitons in the course of the measurements and investigate dynamics of the system. In the weak interaction limit, the system follows the mean-field prediction for a short period of time. Long time evolution reveals many-body effects that are related to an increasing uncertainty of soliton positions. In the strong interaction regime particles behave like impenetrable bosons. Then, the probability densities in the configuration space become identical to the probabilities of noninteracting fermions but the wave functions themselves remember the original Bose statistics. Especially, the phase flips that are key signatures of the solitons in the weak interaction limit, can be observed in the time evolution of the strongly interacting bosons.

arXiv:1605.08211•Physical Review A•Aug 18, 2016•27 citations

Correspondence between dark solitons and the type II excitations of the Lieb-Liniger model

Tomasz Karpiuk, Tomasz Sowiński, Mariusz Gajda +2 more

A one-dimensional model of bosons with repulsive short-range interactions, solved analytically by Lieb and Liniger many years ago, predicts the existence of two branches of elementary excitations. One of them represents Bogoliubov phonons; the other, as suggested by some authors, might be related to dark solitons. On the other hand, it has been already demonstrated within the framework of the classical field approximation that a quasi-one-dimensional interacting Bose gas at equilibrium exhibits excitations which are phonons and dark solitons. By showing that for lower temperatures statistical distributions of dark solitons obtained within the classical field approximation are well represented by the distributions of quasiparticles of the second kind derived from fully quantum description, we demonstrate that type II excitations in the Lieb-Liniger model are, indeed, quantum solitons.

arXiv:1402.5650•Physical Review A•Jan 21, 2015•26 citations
Abstract The coordinate Bethe ansatz can be extended to a model, the Lieb–Liniger model, of a one-dimensional gas of Bosons interacting with repulsive δ-function potentials. It has attracted attention due to its relevance for experimental developments in the fields of ultracold gases and optical lattices. This chapter provides an exposition of the related classical nonlinear Schrödinger equation, followed by its generalization to the quantum model. It explores a limiting case, the Tonks-Girardeau gas. The δ-function potentials supply a kind of boundary condition on the wave functions allowing us to analyze the eigenfunctions of the Bethe ansatz, which are examined on the infinite line and for periodic boundary conditions. The latter leads to the Bethe ansatz equations. The solution of these equations is achieved in the thermodynamic limit for the ground state and for low-lying excited states.
Describing properties of a strongly interacting quantum many-body system poses a serious challenge both for theory and experiment. In this work, we study excitations of one-dimensional repulsive Bose gas for arbitrary interaction strength using a hydrodynamic approach. We use linearization to study particle (type-I) excitations and numerical minimization to study hole (type-II) excitations. We observe a good agreement between our approach and exact solutions of the Lieb-Liniger model for the particle modes and discrepancies for the hole modes. Therefore, the hydrodynamical equations find to be useful for long-wave structures like phonons and of a limited range of applicability for short-wave ones like narrow solitons. We discuss potential further applications of the method.
Eigenstates of Bose particles with repulsive contact interactions in one-dimensional space with periodic boundary conditions can be found with the help of the Bethe ansatz. The type II excitation spectrum identified by E. H. Lieb, reproduces the dispersion relation of dark solitons in the mean-field approach. The corresponding eigenstates possess translational symmetry which can be broken in measurements of positions of particles. We analyze emergence of single and double solitons in the course of the measurements and investigate dynamics of the system. In the weak interaction limit, the system follows the mean-field prediction for a short period of time. Long time evolution reveals many-body effects that are related to an increasing uncertainty of soliton positions. In the strong interaction regime particles behave like impenetrable bosons. Then, the probability densities in the configuration space become identical to the probabilities of noninteracting fermions but the wave functions themselves remember the original Bose statistics. Especially, the phase flips that are key signatures of the solitons in the weak interaction limit, can be observed in the time evolution of the strongly interacting bosons.
A one-dimensional model of bosons with repulsive short-range interactions, solved analytically by Lieb and Liniger many years ago, predicts the existence of two branches of elementary excitations. One of them represents Bogoliubov phonons; the other, as suggested by some authors, might be related to dark solitons. On the other hand, it has been already demonstrated within the framework of the classical field approximation that a quasi-one-dimensional interacting Bose gas at equilibrium exhibits excitations which are phonons and dark solitons. By showing that for lower temperatures statistical distributions of dark solitons obtained within the classical field approximation are well represented by the distributions of quasiparticles of the second kind derived from fully quantum description, we demonstrate that type II excitations in the Lieb-Liniger model are, indeed, quantum solitons.