Self-similarity and attraction in stochastic nonlinear reaction-diffusion systems

Abstract

Self-similarity solutions play an important role in many fields of science. We explore self-similarity in some stochastic partial differential equations (spdes). Important issues are not only the existence of stochastic self-similarity but also whether a self-similar solution is dynamically attractive, and if it is, to what particular solution does the system evolve. By recasting a class of spdes in a form to which stochastic center manifold theory may be applied, we resolve these issues in this class. For definiteness, a first example of self-similarity in solutions of Burgers equation driven by some stochastic force is studied. Under suitable assumptions a stationary solution is constructed which yields the existence of a stochastically self-similar solution for the stochastic Burgers equation. Further, the asymptotic convergence to the self-similar solution is proved. Second, in more general stochastic reaction-diffusion systems, stochastic center manifold theory provides a framework for constructing stochastic self-similar solutions, confirming their relevance, and determining the correct solution for any compact initial condition. Third, we argue that dynamically moving the effective spatial origin and dynamically distorting time improve the description of the stochastic self-similarity. Finally, stochastic self-similarity in an extremely simple model of turbulent mixing shows how anomalous fluctuations may arise in eddy diffusivities. The techniques and results we discuss should be applicable to a wide range of stochastic self-similarity problems.