Steady-state ensemble for the complex Ginzburg-Landau equation with weak noise

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Abstract
The complex Ginzburg-Landau equation with weak noise, the normal form of the amplitude equation for the order parameter in a spatially distributed system undergoing a continuous Hopf bifurcation, is solved in certain limits for its time-independent probability distribution, which governs the steady state in one spatial dimension. The method used consists of solving the Hamilton-Jacobi equation of the nonequilibrium potential associated with the steady-state distribution. The solution is obtained in the limit of weak spatial diffusion of the order parameter. The nonequilibrium potential serves as a Lyapunov functional for the order-parameter field. We use our result to discuss the Newell-Kuramoto instability and the Eckhaus-Benjamin-Feir instability in one spatial dimension, and to calculate potential barriers of the saddles separating plane-wave attractors. The latter ones provide us with a global measure of stability for these attractors.

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