The simulations reveal several classes of localized structures:
a. Solitonic and Breather Modes:
Stable or oscillatory localized peaks in the energy field,
Maintain integrity over time, often oscillating in amplitude ("breathing"),
Resemble soliton trains in 1D or dissipative solitons in higher dimensions.
b. Energy Bubbles:
Spherical or ellipsoidal energy concentration zones,
Surrounded by low-energy voids, indicating nonlinear trapping,
Analogous to inflating vacuum domains in cosmological inflationary theories.
c. Topological Domains:
Phase-locked regions separated by sharp domain boundaries,
Exhibit persistent contrast and stability, resistant to small perturbations,
Similar in topology to magnetic skyrmions or spin textures.
d. Curvature Spikes:
High Laplacian peaks of the energy density interpreted as scalar curvature,
Cluster near regions of nonlinear energy focusing,
Offer analogs to compactified energy zones, black hole cores, or proto-galaxies.
3. Metrics for Structure Detection
We utilize a combination of physical observables to quantify these structures:
Local Energy Density: (x,t)=I(x,t)2\rho(x,t) = |I(x,t)|^2(x,t)=I(x,t)2, to identify localized energy wells.
Phase Gradient: arg(I)\nabla \arg(I)arg(I), to detect domain boundaries and phase defects.
Curvature Proxy: R(x,t)2(x,t)R(x,t) \sim -\nabla^2 \rho(x,t)R(x,t)2(x,t), as an emergent metric field.
Entropy-like Measure: S(t)=(x,t)log(x,t)dxS(t) = \int \rho(x,t) \log \rho(x,t) dxS(t)=(x,t)log(x,t)dx, to track information ordering.
These metrics evolve non-trivially and display clear transitions from disorder (high entropy, homogeneous density) to order (low entropy, structured localizations).
4. Structure Interactions and Evolution
The structures are not static; they interact dynamically through:
Annihilation: Opposite phase or anti-symmetric structures cancel out.
Fusion: Two neighboring localized peaks merge into a stronger excitation.
Repulsion and Orbiting: Certain breather pairs form stable bound states, orbiting each other under effective field-mediated interaction.
These interactions mimic field-theoretic particle dynamics and support the hypothesis that geometry and structure can emerge from pure field dynamics.
