Localized structures in two-field systems: exact solutions in the presence of Lorentz symmetry breaking and explicit connection with geometric constraints

TL;DR

This study analyzes two real scalar fields in (1+1)D with a Lorentz-violating coupling to derive exact static solutions via a Bogomol'nyi-like first-order formalism. It reveals a direct link between Lorentz symmetry breaking and geometric constraints previously observed in geometrically constrained domain-wall systems, and systematically builds three model families that either reproduce those constrained solutions or uncover new internal structures, such as lump-like profiles and regions of negative energy density. The results broaden the understanding of how Lorentz-violating terms shape topological defects and suggest practical implications for condensed matter domains and broader nonlinear systems, with potential future work on soliton collisions, spectral properties, and higher-field generalizations. The constructed framework provides explicit analytic solutions and coordinates that encode geometric constraints within Lorentz-violating dynamics, offering a bridge between field theory and constrained-material phenomena.

Abstract

We investigate a class of models described by two real scalar fields in two-dimensional spacetime. The study focuses mainly on the presence of exact static solutions which satisfy the first-order formalism, in models constructed to engender Lorentz symmetry violation. We start by exploring a direct connection between Lorentz breaking and geometric constraint, as experimentally examined in the case of domain walls in geometrically constrained magnetic materials. By means of a specific choice of functions, we show that imposing geometric constraint within the Lorentz-violating framework recovers the exact solutions of the corresponding Lorentz-invariant theory. Furthermore, we extend the investigation to new models that go beyond reproducing the Lorentz invariant geometrically constrained solutions, revealing that it remains possible to parametrize the first-order equation of one of the fields through a suitably redefined coordinate.

Figures (9)

• Figure 1: Top view of the potential \ref{['Pot1']} in the $(\phi,\chi)$ plane.
• Figure 2: Top view of the potential \ref{['Pot2']} in the $(\phi,\chi)$ plane, for $n=2$.
• Figure 3: Top view of the potential \ref{['Pot3']} in the $(\phi,\chi)$ plane, for $\alpha=1$ and $b=0.4$.
• Figure 4: The solutions for $\phi(x)$\ref{['V']} and $\chi(x)$\ref{['CV']} are shown for $\alpha=1$, with the dash-dotted black line representing $\phi(x)$. We depict $\chi(x)$ for $b= \pm0.2, \pm0.4, \pm0.8, \pm1$, represented by blue, red, yellow and purple lines, respectively. The top and bottom panels refer to positive and negative values of $b$, respectively.
• Figure 5: Top view of the potential \ref{['Pot4']} in the $(\phi,\chi)$ plane, for $\alpha=-1$ and $b=-0.4$ (top) and for $\alpha=1$ and $b=0.4$ (bottom).