Complex Ginzburg-Landau equation — gain, loss, and self-sustained pulses
Complex Ginzburg-Landau equation (CGLE):
∂A/∂t = δA + (1+iβ)∂²A/∂x² + (ε+iν)|A|²A + μ|A|⁴A
describes the amplitude of an oscillatory field near a Hopf bifurcation. Unlike conservative NLS solitons, dissipative solitons are fixed-shape attractors maintained by a balance between gain (ε, at low intensity) and loss (μ, at high intensity). The linear gain δ<0 keeps the background stable while cubic gain ε>0 amplifies the pulse. Adjust parameters to watch solitons form, breathe, or dissolve.