2. Energy Bubbles in 2D/3D
In higher dimensions, localized energy density can form "bubbles"---regions of high I2|I|^2I2 bounded by lower-amplitude surroundings. These bubbles:
Exhibit quasi-static behavior for long timescales,
Act as localized curvature spikes in the emergent metric interpretation,
Can attract or repel other propagating excitations, akin to gravitational lensing.
In 3D, they resemble information condensates, with field energy trapped in a spherical volume due to a balance of dispersive and nonlinear effects.
3. Curvature Spikes and Singular Lensing Regions
As I(x,t)2|I(\vec{x}, t)|^2I(x,t)2 increases, the effective refractive index neff1+I2n_{\text{eff}} \sim 1 + \lambda |I|^2neff1+I2 leads to local refractive gradients, bending phase fronts and forming converging or diverging zones. These manifest as:
Curvature spikes, i.e., localized increases in scalar curvature (analog),
Effective "proto-singularities", where energy density approaches theoretical limits,
Trapping regions or analog black-hole-like behavior if gradient and damping allow.
4. Topological Quantization and Stability
When boundary conditions and system geometry allow, these structures may become topologically protected. Examples include:
Vortex rings and spin-like textures in phase space,
Winding number--protected solitons, especially in closed 2D manifolds,
Skyrmion-like configurations, where vectorial or multicomponent III is used.
Such features resist perturbation, serving as robust carriers of information topology and modeling early universe features like cosmic strings or domain walls.
5. Dynamical Formation and Annihilation
Numerical simulations and perturbation analysis suggest that:
Excitation bursts above a critical threshold can nucleate soliton-bubble states.
Collisions between two such structures may lead to mergers, phase shifts, or annihilation, mimicking cosmological inflation bubbles or early-universe phase transitions.
Energy can temporarily localize, then disperse in a chaos-to-order cycle, potentially linked to symmetry breaking and spontaneous structure emergence.
6. Implications for Analog Spacetime Engineering
