[Paper Review] Deformed solitons: The case of two coupled scalar fields

A. de Souza Dutra|arXiv (Cornell University)|May 22, 2007

Nonlinear Waves and Solitons3 citations

TL;DR

This paper presents a generalized method to generate new exact solitonic models in 1+1 dimensions from known two-coupled scalar field systems by deforming the fields via nonlinear transformations. The key contribution is a systematic procedure to restore the BPS property—ensuring first-order equations yield valid second-order solutions—by introducing correction functions to the superpotential, enabling the construction of deformed BPS solitons with extended topological structures beyond the original model.

ABSTRACT

In this work, we present a general procedure, which is able to generate new exact solitonic models in 1+1 dimensions, from a known one, consisting of two coupled scalar fields. An interesting consequence of the method, is that of the appearing of nontrivial extensions, where the deformed systems presents other BPS solitons than that appearing in the original model. Finally we take a particular example, in order to check the above mentioned features.

Motivation & Objective

• To address the failure of naive field transformations in preserving BPS conditions for coupled scalar field systems.
• To develop a general procedure for generating new exact solitonic models from known ones in 1+1 dimensions with two coupled scalar fields.
• To restore the BPS property—specifically the symmetry of mixed partial derivatives of the superpotential—after field deformation.
• To construct deformed systems that support new BPS solitons not present in the original model.
• To extend the framework for potential applications in topological defects, soliton networks, and quantum field theories.

Proposed method

• Apply a nonlinear field transformation: φ = f(θ, φ), χ = g(θ, φ), mapping the original fields to new variables.
• Derive first-order differential equations in the new variables using the chain rule and Jacobian determinant J(θ, φ).
• Identify the inconsistency in the superpotential’s mixed derivatives (Wθφ ≠ Wφθ), which breaks the BPS condition.
• Introduce arbitrary correction functions H₁(φ) and H₂(θ) to restore Wθφ = Wφθ symmetry.
• Fix the correction functions by using the known solution relations between original and transformed fields (e.g., θ(φ) or φ(θ)).
• Reconstruct the consistent superpotential W(θ, φ) and derive the corrected BPS equations with proper second-order consistency.

Experimental results

Research questions

• RQ1 Can a naive field transformation preserve the BPS property in two coupled scalar field systems?
• RQ2 What conditions must be imposed on the superpotential to ensure consistency between first- and second-order equations after field deformation?
• RQ3 Can new BPS solitons emerge in the deformed system that are absent in the original model?
• RQ4 How can arbitrary functions be determined to restore the symmetry of mixed partial derivatives in the superpotential?
• RQ5 What is the structure of the deformed potential and soliton solutions in the corrected system?

Key findings

• The naive field transformation fails to preserve the BPS condition due to Wθφ ≠ Wφθ, invalidating the first-order solutions as solutions to the second-order equations.
• By introducing a correction function H₁(φ) = ∫dφ (sinh²φ − a²)[2μ sechφ + (λ − 2μ) coshφ], the symmetry Wθφ = Wφθ is restored.
• The corrected system yields a consistent superpotential W₁(θ, φ) = μ sinhφ θ² + H₁(φ), ensuring the BPS property is preserved.
• The deformed system supports new BPS solitons, including extended configurations not present in the original model.
• The method successfully generates a new solitonic model with a well-defined potential and exact soliton solutions, verified by recovering the original soliton profiles in the transformed variables.
• The framework is extendable to non-BPS states, multi-field systems, and topological defect networks.