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International Journal of Modern Physics B

Localized scalar field configurations in a logarithmic Klein–Gordon field

Kevin Hernández, Elías Castellanos

Published Sep 10, 2026• 0 citations
Publisher

Abstract

We investigate localized, self-bound scalar-field configurations in relativistic Bose–Einstein condensates described by a nonlinear Klein–Gordon equation with cubic and logarithmic interactions. Working in the condensate rest frame, we formulate the corresponding relativistic field theory, derive the conserved Noether current and energy–momentum tensor, and obtain the relativistic chemical potential governing self-confinement, which reduces to a generalized logarithmic Gross–Pitaevskii equation in the nonrelativistic limit. Using a Gaussian variational approach, the field dynamics are reduced to a single collective coordinate describing the condensate width. We derive an exact cubic equation for the equilibrium radius and establish a quantitative stability criterion demonstrating that the logarithmic interaction, rather than the relativistic mass term, provides the restoring mechanism responsible for confinement. Numerical simulations for 7 Li, [Formula: see text]Rb and [Formula: see text]Na confirm dynamically stable, localized configurations exhibiting bounded breathing oscillations within the predicted stability regime, with an equilibrium radius that scales linearly with particle number. These self-confined states display the characteristic dynamical signatures of droplet-like configurations, suggesting that logarithmic nonlinearities provide an alternative mechanism for the formation of localized condensates beyond the conventional Lee–Huang–Yang stabilization scenario.

DOI
10.1142/s0217979226502711

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We investigate localized, self-bound scalar-field configurations in relativistic Bose–Einstein condensates described by a nonlinear Klein–Gordon equation with cubic and logarithmic interactions. Working in the condensate rest frame, we formulate the corresponding relativistic field theory, derive the conserved Noether current and energy–momentum tensor, and obtain the relativistic chemical potential governing self-confinement, which reduces to a generalized logarithmic Gross–Pitaevskii equation in the nonrelativistic limit. Using a Gaussian variational approach, the field dynamics are reduced to a single collective coordinate describing the condensate width. We derive an exact cubic equation for the equilibrium radius and establish a quantitative stability criterion demonstrating that the logarithmic interaction, rather than the relativistic mass term, provides the restoring mechanism responsible for confinement. Numerical simulations for 7 Li, [Formula: see text]Rb and [Formula: see text]Na confirm dynamically stable, localized configurations exhibiting bounded breathing oscillations within the predicted stability regime, with an equilibrium radius that scales linearly with particle number. These self-confined states display the characteristic dynamical signatures of droplet-like configurations, suggesting that logarithmic nonlinearities provide an alternative mechanism for the formation of localized condensates beyond the conventional Lee–Huang–Yang stabilization scenario.
We study a relativistic scalar field model for self-bound Bose-Einstein condensates (BECs) by analyzing a nonlinear Klein-Gordon equation with cubic and logarithmic interactions. This framework captures essential features of quantum droplets, such as self-trapping and finite energy configurations, which emerge from the interplay between attractive and repulsive terms. By performing the non-relativistic limit, we derive a generalized Gross-Pitaevskii equation with a logarithmic correction, consistent with recent models used to describe ultra-cold atomic gasses beyond mean-field theory. We construct the corresponding Lagrangian density, identify conserved quantities via Noether's theorem, and compute the energy-momentum tensor. Numerical solutions of the BEC parameters are shown, establishing the foundations for a field theoretical description of relativistic condensates with a logarithmic interaction. This model provides a unified approach to investigate relativistic effects in quantum droplets and enriches the theoretical landscape of Bose-Einstein condensates with non-standard interactions. The resulting dynamics exhibit stable oscillatory regimes consistent with self-bound condensate configurations.
The nonlinear dynamics of self-bound Bose–Einstein matter waves under the action of an attractive nonlocal and a repulsive local interaction is analyzed by means of a time-dependent variational formalism. The mean-field model described by the Gross–Pitaevskii equation (GPE) is reduced to a single second-order conservative ordinary differential equation admitting a potential function. The stable fixed points and the linear oscillation frequencies of the breather modes are determined. The chosen nonlocal interactions are a Gaussian-shaped potential and a van der Waals-like potential. The variational solutions are compared with direct numerical simulations of the GPE, for each of these nonlocalities. The spatio-temporal frequency spectra of linear waves are also determined.
Abstract We study the formation of a stable self-trapped spherical quantum ball in a binary Bose–Einstein condensate (BEC) with two-body inter-species attraction and intra-species repulsion employing the beyond-mean-field Lee–Huang–Yang and the three-body interactions. We find that either of these interactions or a combination of them can stabilize the binary BEC quantum ball with very similar stationary results, and for a complete description of the problem both the terms should be considered. These interactions lead to higher-order nonlinearities, e.g. quartic and quintic, respectively, in a nonlinear dynamical equation compared to the cubic nonlinearity of the two-body contact interaction in the mean-field Gross–Pitaevskii equation. The higher-order nonlinearity makes the energy infinitely large at the center of the binary ball and thus avoids its collapse. In addition to the formation of stationary binary balls, we also study a collision between two such balls. At large velocities, the collision is found to be elastic, which turns out to be inelastic as the velocity is lowered. We consider the numerical solution of a beyond-mean-field model for the binary ball as well as a single-mode variational approximation to it in this study.
Using variational and numerical solutions of the mean-field Gross-Pitaevskii equation for attractive interaction (with cubic or Kerr nonlinearity) we show that a stable bound state can appear in a Bose-Einstein condensate (BEC) in a localized exponentially-screened radially-symmetric harmonic potential well in two and three dimensions. We also consider an axially-symmetric configuration with zero axial trap and a exponentially-screened radial trap so that the resulting bound state can freely move along the axial direction like a soliton. The binding of the present states in shallow wells is mostly due to the nonlinear interaction with the trap playing a minor role. Hence these BEC states are more suitable to study the effect of the nonlinear force on the dynamics. We illustrate the highly nonlinear nature of breathing oscillation of these states. Such bound states could be created in BECs and studied in the laboratory with present knowhow.
We present the theoretical and mathematical foundations of stability analysis for a Bose–Einstein condensate (BEC) at Dirac points of a honeycomb optical lattice. The combination of s -wave scattering for bosons and lattice interaction places constraints on the mean-field description, and hence on vortex configurations in the Bloch-envelope function near the Dirac point. A full derivation of the relativistic linear stability equations (RLSE) is presented by two independent methods to ensure veracity of our results. Solutions of the RLSE are used to compute fluctuations and lifetimes of vortex solutions of the nonlinear Dirac equation, which include Anderson–Toulouse skyrmions with lifetime s. Beyond vortex stabilities the RLSE provide insight into the character of collective superfluid excitations, which we find to encode several established theories of physics. In particular, the RLSE reduce to the Andreev equations, in the nonrelativistic and semiclassical limits, the Majorana equation, inside vortex cores, and the Dirac–Bogoliubov–de Gennes equations, when nearest-neighbor interactions are included. Furthermore, by tuning a mass gap, relative strengths of various spinor couplings, for the small and large quasiparticle momentum regimes, we obtain weak-strong Bardeen–Cooper–Schrieffer superconductivity, as well as fundamental wave equations such as Schrödinger, Dirac, Klein–Gordon, and Bogoliubov–de Gennes equations. Our results apply equally to a strongly spin–orbit coupled BEC in which the Laplacian contribution can be neglected.
Abstract. In the infinite three-dimensional space, we study a homogeneous nonrelativistic Schrödinger Hamiltonian describing a self-gravitating Bose gas with two types of interactions: a repulsive scaled interaction modeled by [Formula: see text] and an attractive Newtonian potential of the form [Formula: see text]. We investigate the mean-field limit as the number of particles [Formula: see text], with coupling parameters [Formula: see text] and [Formula: see text] satisfying [Formula: see text]. In this regime, we rigorously derive the corresponding Gross–Pitaevskii–Poisson theory. Furthermore, we analyze the asymptotic behavior of the condensate profile in two opposite limiting cases: when [Formula: see text], the system is well approximated by a Thomas–Fermi-type model, whereas when [Formula: see text], the condensate approaches a Choquard–Pekar-type profile.