Appendix II. Stability Maps and Bifurcation Diagrams
1. Overview and Purpose
The nonlinear field dynamics governed by the equation:
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)
exhibit rich behaviors, including pattern formation, localized excitations, oscillons, and topological phase defects. This appendix provides stability maps and bifurcation diagrams to classify the system's response under variation of key parameters such as:
Nonlinearity strength \lambda,
Damping \gamma,
External forcing amplitude and shape B(x,t)B(x,t)B(x,t),
Initial conditions (pulse width, amplitude, and phase distribution).
2. Methodology
We explore the parameter space via:
a. Linear Stability Analysis
For small perturbations around the trivial solution I=0I = 0I=0, assume:
I(x,t)=ei(kxt),1I(x,t) = \epsilon e^{i(kx - \omega t)},\quad \epsilon \ll 1I(x,t)=ei(kxt),1
Substitute into the linearized equation:
