where \kappa is an analog of the Einstein constant, controlling the coupling between the emergent energy density and curvature.
In 2D/3D simulations, regions with self-trapped pulses or solitonic cores show phase-front bending, interference pattern deformation, and signal redirection --- behaviors that mirror geodesic deviation in curved spacetime.
Localized Structures and Proto-Geometry
Excitations in higher dimensions can stabilize into:
Vortex-like configurations (in 2D), analogous to cosmic strings.
Breathers and skyrmions, depending on phase coherence.
Localized 'traps' for signal pulses, resembling black-hole analogs in effective refractive index.
These patterns obey topological constraints and can be associated with winding numbers or homotopy classes, establishing a link to quantized curvature or discrete topology transitions.
Metric Backreaction and Self-Modulation
Interestingly, the evolution of III is both shaped by and shapes the effective metric. This bidirectional feedback---a hallmark of gravitational systems---is captured here in a self-modulating nonlinear wave equation where field intensity governs its own dispersion landscape.
For example, the effective refractive index becomes:
neff(x,t)1c2[1+I(x,t)2]n_{\text{eff}}(\vec{x}, t) \sim \frac{1}{c^2} \left[ 1 + \lambda |I(\vec{x}, t)|^2 \right]neff(x,t)c21[1+I(x,t)2]
This causes wavefront bending, trapping, or focusing, directly tied to the energy localization.
Implications for Analog Cosmology
