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6. Summary of Critical Thresholds (Example Values)
Values depend on simulation grid size and boundary conditions.
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Appendix III: Simulation Codes and Benchmarking Comparisons
1. Codebase Architecture
The simulation framework was developed using Python 3.11 with optimized numerical libraries. The core solvers are based on the split-step Fourier method (SSFM) and pseudo-spectral techniques for handling the nonlinear partial differential equation (PDE):
I+Ic22I+I2I=B(x,t)\ddot{I} + \gamma \dot{I} - c^2 \nabla^2 I + \lambda |I|^2 I = B(x,t)I+Ic22I+I2I=B(x,t)
This PDE is solved on a 2D or 3D periodic grid, discretized as follows:
Time integration: Runge--Kutta 4 or symplectic integrators.
Spatial derivatives: FFT-based spectral differentiation.
Nonlinear term: Treated explicitly in real space.
Boundary conditions: Periodic or absorbing (via damping mask).
Libraries used:
numpy, scipy, matplotlib
pyFFTW (for accelerated Fourier transforms)
numba (for JIT compilation of critical loops)
h5py (for saving large 3D datasets)
2. Simulation Parameters (Benchmark Case)
