Topological solitons of two-field scalar theories in rotationally symmetric backgrounds

Abstract: This work concerns scalar field theories with topologically nontrivial vacuum manifold in rotationally symmetric backgrounds of arbitrary dimension. Lagrangians with canonical and generalized kinetic terms are considered, and a Bogomol'nyi framework is developed for the symmetric restriction of the theory. Localized topological solutions are found. Their stability, which would normally be prevented in higher dimensions due to scaling instability, is made possible by the presence of an explicit radial dependence on the potential. The first-order equations give rise to an integrable orbit equation which can be used to solve the problem completely. It is shown that target space orbits - but not the solutions themselves - are shared between analogous systems defined in different backgrounds. Moreover, the first-order equations can be mapped into a one-dimensional BPS theory through a transformation encoded by a function $ξ(r)$. The internal structure, size and existence of defects follows from the properties and range of this mapping. We use these tools to evaluate the effect of geometry on confinement, existence, and structure of solitons. Exact solutions are provided in Minkowski, Schwarzschild, de Sitter, Schwarzschild de Sitter and conformally flat backgrounds.

• The paper introduces a novel BPS framework that leverages radial-dependent potentials to stabilize two-field topological solitons across diverse geometries.
• It employs integrable orbit equations and variable transformations to derive exact solutions in both canonical BNRT and generalized kinetic models.
• Results demonstrate geometry-induced modulation of soliton confinement, offering scalable strategies for applications in brane physics and cosmology.

Topological Solitons in Two-Field Scalar Theories with Rotational Symmetry

Overview

This paper investigates the structure, existence, and stability of topological solitons in scalar field theories with two fields, formulated in rotationally symmetric backgrounds of arbitrary dimension. The authors extend Bogomol'nyi–Prasad–Sommerfield (BPS) techniques to both canonical and generalized kinetic Lagrangians, demonstrating how explicit radial dependence in the scalar potential enables localized, stable solutions in dimensions where Derrick's theorem would otherwise forbid them. The approach systematically links the effect of geometry to the soliton's internal structure via an integrable orbital mapping, yielding exact solutions in Minkowski, Schwarzschild, de Sitter, Schwarzschild–de Sitter, and conformally flat spacetimes.

Theoretical Framework

The paper begins by establishing the formalism for one-field BPS theories and generalizes to two-field Lagrangians of the form:

$$S = \int d^{D+1}x\,\sqrt{|g|}\left[\frac{P(\phi,\chi)}{2}g^{\mu\nu} \partial_\mu\phi \partial_\nu\phi + \frac{Q(\phi,\chi)}{2}g^{\mu\nu} \partial_\mu\chi \partial_\nu\chi - V(\phi,\chi,r) \right].$$

The authors invoke a metric ansatz for static, rotationally symmetric geometries:

$$ds^2 = A^2(r)dt^2 - [B^2(r)dr^2 + \rho^2(r)d\Omega_{D}^2],$$

with explicit computation of the Jacobian $\gamma(r)$, thus accommodating spacetimes of arbitrary curvature and dimensionality.

A key innovation is the use of explicitly radial-dependent potentials of the form:

$$V(\phi,\chi,r)=\frac{B^2(r)}{2\gamma^2(r)}\left(\frac{W_\phi^2}{P(\phi,\chi)} + \frac{W_\chi^2}{Q(\phi,\chi)}\right),$$

where $W(\phi,\chi)$ is an auxiliary superpotential. The explicit $r$ dependence breaks translation invariance but introduces a mechanism to stabilize defects against Derrick-type scaling instabilities.

BPS Construction and Orbit Structure

The corresponding Bogomol'nyi bound admits first-order equations for static solutions: $$\phi'(r)=\pm \frac{W_\phi B^2(r)}{\gamma(r) P}, \quad \chi'(r)=\pm \frac{W_\chi B^2(r)}{\gamma(r) Q}.$$ The orbit equation, central to the classification of defects in field-theoretic models, becomes:

$\frac{d\phi}{d\chi} = \frac{W_\phi Q}{W_\chi P},$

which can often be reduced to an integrable form given an appropriate integrating factor. The remarkable result is that the target space orbits are geometry-independent, although the field profiles themselves are background-dependent.

A variable transformation defined by

$\frac{d\xi(r)}{dr} = \frac{B^2(r)}{\gamma(r)}$

maps the system into an effective one-dimensional BPS theory, facilitating solution classification and boundary condition transfer across different geometries. This approach unifies soliton analysis in diverse backgrounds via a single parameter $\xi(r)$.

Explicit Models and Analytical Solutions

Canonical Two-Field Models (BNRT Type)

The BNRT model, specified by $$ds^2 = A^2(r)dt^2 - [B^2(r)dr^2 + \rho^2(r)d\Omega_{D}^2],$$0, demonstrates how exact soliton solutions can be constructed and analyzed in both flat and curved backgrounds. The vacuum structure $$ds^2 = A^2(r)dt^2 - [B^2(r)dr^2 + \rho^2(r)d\Omega_{D}^2],$$1 allows for rich orbit topology.

A strong result is the demonstration that in $$ds^2 = A^2(r)dt^2 - [B^2(r)dr^2 + \rho^2(r)d\Omega_{D}^2],$$2, asymptotically flat backgrounds do not support nontrivial solutions due to the boundedness of $$ds^2 = A^2(r)dt^2 - [B^2(r)dr^2 + \rho^2(r)d\Omega_{D}^2],$$3, whereas spacetimes with unbounded $$ds^2 = A^2(r)dt^2 - [B^2(r)dr^2 + \rho^2(r)d\Omega_{D}^2],$$4 (e.g., pure de Sitter or conformally deformed backgrounds) allow for soliton existence and geometry-induced modification of the defect's spatial extent.

Generalized Kinetic Models

Generalized kinetic terms ($$ds^2 = A^2(r)dt^2 - [B^2(r)dr^2 + \rho^2(r)d\Omega_{D}^2],$$5, $$ds^2 = A^2(r)dt^2 - [B^2(r)dr^2 + \rho^2(r)d\Omega_{D}^2],$$6 functions) enable the construction of separable models where $$ds^2 = A^2(r)dt^2 - [B^2(r)dr^2 + \rho^2(r)d\Omega_{D}^2],$$7 and $$ds^2 = A^2(r)dt^2 - [B^2(r)dr^2 + \rho^2(r)d\Omega_{D}^2],$$8 can be solved in sequence. The interplay of kinetic term degeneracy and background geometry produces compacton-like profiles, with soliton sizes and internal structure adjustable via floor parameters in the kinetic couplings.

For higher $$ds^2 = A^2(r)dt^2 - [B^2(r)dr^2 + \rho^2(r)d\Omega_{D}^2],$$9-models (e.g., $\gamma(r)$0 higher powers of $\gamma(r)$1), compact solutions are shown to exist in dimensions and backgrounds where canonical models cannot produce localized defects. The solutions are explicit, parameterized by moduli ($\gamma(r)$2, $\gamma(r)$3), and demonstrate tunable confinement and defect size as direct functions of geometric and dynamical variables.

Mixed BNRT/p-Model Hybrids

By incorporating nontrivial $\gamma(r)$4 and $\gamma(r)$5 functions and constructing potentials with combined BNRT and $\gamma(r)$6-model characteristics, the authors establish new families of integrable orbit equations, yielding field profiles with precise compactification radii in Schwarzschild and other backgrounds. The ability to confine solitons to arbitrarily small or large regions further enhances model applicability, especially for brane and astrophysical system construction.

Implications and Future Directions

The findings have several broad theoretical and practical implications:

• Geometry–Soliton Coupling: The mapping of geometry effects onto soliton profiles via $\gamma(r)$7 provides a rigorous foundation for analyzing topological defects in arbitrary (static, rotationally symmetric) backgrounds, including black hole and cosmological spacetimes.
• Stable Defect Construction: The explicit radial dependence in potentials offers a loophole around Derrick's theorem, permitting finite energy, stable defects in higher dimensions and nonflat backgrounds.
• Analytical Solution Techniques: The integrable orbit construction enables systematic solution generation for a wide variety of models, including those with elaborate kinetic structures.
• Applications in Brane Physics and Cosmology: The extension to thick brane configurations and the flexibility in defect size modulation are directly relevant for stabilized extra-dimensional models and potential dark matter structures.
• Generalization to Dynamical and Non-BPS Systems: Although the present analysis is static, the mathematical apparatus can underpin future investigations of dynamical soliton scattering, excitation spectra, and coupling to gauge fields.

Further research should explore dynamical properties, stability analysis in full nonlinear evolution, and extension to theories with higher-derivative (K-essence) or non-Abelian structure. The orbit equation integrability in mixed-model scenarios deserves closer scrutiny, as does the coupling to quantum gravity candidates and supersymmetric extensions.

Conclusion

This paper rigorously establishes a generalized framework for constructing and analyzing topological solitons in two-field scalar theories on rotationally symmetric backgrounds of arbitrary dimension (2605.14160). By leveraging explicit coordinate-dependent potentials and integrable orbit equations, it achieves stable, localized defect solutions in both canonical and generalized kinetic contexts. The geometric mapping via $\gamma(r)$8 unifies the analysis across diverse spacetimes and enables precise control of soliton confinement and internal structure, with practical implications for brane modeling, cosmological phase transitions, and studies of nonperturbative effects in high-energy physics.