Self-Similarity and Attraction in Stochastic Nonlinear Reaction-Diffusion Systems

Similarity solutions play an important role in many fields of science: we consider here similarity in stochastic dynamics. Important issues are not only the existence of stochastic similarity, but also whether a similarity solution is dynamically attractive, and if it is, to what particular solution does the system evolve. By recasting a class of stochastic PDEs in a form to which stochastic centre manifold theory may be applied we resolve these issues in this class. For definiteness, a first example of self-similarity of the Burgers’ equation driven by some stochastic forced is studied. Under suitable assumptions, a stationary solution is constructed which yields the existence of a stochastic self-similar solution for the stochastic Burgers’ equation. Furthermore, the asymptotic convergence to the self-similar solution is proved. Second, in more general stochastic reaction-diffusion systems stochastic centre manifold theory provides a framework to construct the similarity solution, confirm its relevance, and determines the correct solution for any compact initial condition. Third, we argue that dynamically moving the spatial origin and dynamically stretching time improves the description of the stochastic similarity. Lastly, an application to 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 similarity problems. ∗School of Mathematics, University of Adelaide, South Australia, Australia. mailto: [email protected]; and Department of Mathematics, Nanjing University, Nanjing, China. mailto:[email protected] †School of Mathematics, University of Adelaide, South Australia, Australia. mailto: [email protected] 1 ar X iv :1 11 1. 13 71 v1 [ m at h. D S] 6 N ov 2 01 1