I+Ic22I=0=i2c2k2(2)2\ddot{I} + \gamma \dot{I} - c^2 \nabla^2 I = 0 \Rightarrow \omega = -i\frac{\gamma}{2} \pm \sqrt{c^2 k^2 - \left( \frac{\gamma}{2} \right)^2}I+Ic22I=0=i2c2k2(2)2
Instability condition: c2k2>2/4c^2 k^2 > \gamma^2 / 4c2k2>2/4
b. Numerical Spectral Stability Analysis
For nonlinear steady states Is(x)I_s(x)Is(x), we perform:
I(x,t)=Is(x)+(x,t),1I(x,t) = I_s(x) + \delta(x,t), \quad \delta \ll 1I(x,t)=Is(x)+(x,t),1
and compute the Floquet or Lyapunov spectra to assess growth rates of perturbations.
c. Bifurcation Tracking
We apply continuation methods to trace fixed points, their stability, and bifurcation points as a function of a control parameter (e.g., \lambda, B0B_0B0).
3. Stability Phase Diagrams
We numerically construct stability maps in the (,)(\lambda, \gamma)(,) plane for fixed external excitation profiles. Results show distinct regions:
Stable localized states: Energy bubbles or oscillons remain bounded and coherent.
Breathers: Periodic energy localization and collapse cycles.
Chaotic turbulence: Noisy, nonstationary field with broad frequency content.
Collapse regime: Energy density diverges locally, indicating singular behavior.
Sample Map:
