Out-of-equilibrium quantum fluids (exciton-polariton condensates, cold atoms) exhibit dissipative solitons and vortex lattices. Search for "nonequilibrium quantum pattern formation" on arXiv.
For oscillatory media (e.g., chemical oscillations or superconductors), the CGLE describes near-threshold dynamics: [ \frac\partial A\partial t = A + (1 + i\alpha)\nabla^2 A - (1 + i\beta)|A|^2 A ] Solutions include plane waves, spiral defects, and spatiotemporal chaos. pattern formation and dynamics in nonequilibrium systems pdf
Cambridge University Press.
Close to a bifurcation point, the slow evolution of pattern amplitude is described by universal equations such as the Ginzburg-Landau equation (for stationary patterns) or the Complex Ginzburg-Landau equation (for oscillatory patterns). A PDF of Cross & Hohenberg’s "Pattern Formation Outside of Equilibrium" (Reviews of Modern Physics, 1993) is the gold standard here. For oscillatory media (e
A generic two-species reaction-diffusion system: where f, g describe local reactions, and D_u,
∂u/∂t = D_u ∇²u + f(u,v)
∂v/∂t = D_v ∇²v + g(u,v)
where f, g describe local reactions, and D_u, D_v are diffusion coefficients.
The theoretical backbone of pattern formation is found in nonlinear partial differential equations. While the specifics vary, the emergence of patterns usually follows a universal pathway.
