Pattern Selection: Modes that match the system's natural nonlinear resonance persist. These include ring-shaped wavefronts, spirals, or bubble-like domains.
Solitonic Structures: In 1D, these manifest as stable moving pulses; in 2D/3D, as breathing bubbles, vortex pairs, or quasi-stationary lumps.
Topological Features: Some simulations reveal phase defects, vorticity lines, or energy density domains that resemble curvature spikes, suggesting topological excitation analogs.
Energy becomes spatially localized and temporally trapped, enabling a geometric interpretation of evolving regions as curvature analogs.
3. Long-time Behavior and Meta-stability
As the simulation progresses further:
Persistent Patterns: Certain configurations become long-lived, akin to meta-stable vacua in quantum field theory.
Phase-structured Domains: Multi-modal regions emerge where the phase of III forms coherent domains separated by phase walls or domain boundaries.
Localized Entropy Production: Localized instabilities can form and dissolve, contributing to entropy-like behavior even in conservative systems.
Depending on boundary conditions and nonlinearity \lambda, the system may either:
Relax into a stable, structured state, or
Enter a regime of quasi-periodic or chaotic breathing.
4. Evolution in Higher Dimensions
In 2D simulations, we observe:
Radially symmetric expansion from the blink core,
Formation of energy wells and curvature ridges,
Onset of vortex--antivortex pairs when the phase gradient exceeds a critical value.
In 3D simulations (at reduced spatial resolution due to computational constraints), these effects translate to:
Spherical shell formation, reminiscent of acoustic cosmology analogs,
Emergence of filamentary structures,
Localized "inflation" zones followed by field collapse, suggesting mini big-crunch cycles within a bounded domain.
5. Spectral Evolution and Energy Channeling
To track energy movement across time, we compute the Fourier transform of I(x,t)I(\vec{x}, t)I(x,t):
I~(k,t)=eikxI(x,t)dx\tilde{I}(\vec{k}, t) = \int e^{-i \vec{k} \cdot \vec{x}} I(\vec{x}, t) \, d\vec{x}I~(k,t)=eikxI(x,t)dx
Analysis of I~(k,t)2|\tilde{I}(\vec{k}, t)|^2I~(k,t)2 reveals:
