We systematically varied A0A_0A0, \sigma, and \lambda to scan the nonlinear excitation threshold. Key regimes observed:
Sub-threshold excitations (small A0A_0A0): Pulse dissipates quickly; no emergent structure.
Critical regime: Leads to the formation of oscillatory localized states (breathers).
Super-threshold regime: Triggers soliton formation, expanding bubble domains, and in some cases, spontaneous symmetry breaking.
3. Spatial Dimensionality Considerations
Simulations were run in:
1D to observe soliton propagation and phase locking,
2D to observe radial symmetry breaking and vortex pair creation,
3D (reduced-resolution) to explore spherical bubble collapse, rebound, and curvature concentration.
Blink pulses in 2D and 3D generate radial energy waves, whose interference patterns give rise to self-organized geometries and localized curvature spikes.
4. Pulse Duration and Temporal Profile
In extended versions, time-dependent blinking is added via:
B(x,t)=A(t)exp(xx0222)B(\vec{x}, t) = A(t) \exp\left(-\frac{|\vec{x} - \vec{x}_0|^2}{2\sigma^2}\right)B(x,t)=A(t)exp(22xx02)
With A(t)=A0sech(t)A(t) = A_0 \cdot \text{sech}(\omega t)A(t)=A0sech(t), controlling temporal sharpness. This mimics ultrafast pump-laser pulses or magnetoelastic modulations, depending on the analog platform.
Short 1\omega^{-1}1: Strong spectral broadening, mimicking high-entropy pre-structure epochs.
Longer durations: Promote resonant mode formation and pattern synchronization.
5. Simulation Tools and Boundary Conditions
The simulations are performed using finite-difference time-domain (FDTD) and pseudo-spectral methods, ensuring accurate handling of both dispersion and nonlinearity. Boundary conditions:
Periodic: For topology-sensitive structure formation.
Absorbing: For open-universe analogs, preventing reflection.
Spatial resolution x\Delta xx and temporal step t\Delta tt are chosen to satisfy the Courant-Friedrichs-Lewy (CFL) condition for numerical stability.
