Global potential, topology, and pattern selection in a noisy stabilized Kuramoto–Sivashinsky equation

Abstract

We formulate a general method to extend the decomposition of stochastic dynamics developed by Ao et al. [J. Phys. Math. Gen.37, L25–L30 (2004)] to nonlinear partial differential equations which are nonvariational in nature and construct the global potential or Lyapunov functional for a noisy stabilized Kuramoto–Sivashinsky equation. For values of the control parameter where singly periodic stationary solutions exist, we find a topological network of a web of saddle points of stationary states interconnected by unstable eigenmodes flowing between them. With this topology, a global landscape of the steady states is found. We show how to predict the noise-selected pattern which agrees with those from stochastic simulations. Our formalism and the topology might offer an approach to explore similar systems, such as the Navier Stokes equation.