Localized Scalar Field configurations in a Logarithmic Klein–Gordon Field

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 selfconfinement, which reduces to a generalized logarithmic Gross–Pitaevskii equation in the non-relativistic 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, 87 Rb, and 23 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.