Correlation Moment Analysis and the Time Dependence of Coherence in Systems Described by Nonlinear Partial Differential Equations

Abstract

The work presented introduces correlation moment analysis. This technique can be employed to explore the growth of determinism from stochastic initial conditions in physical systems described by non-linear partial differential equations (PDEs) and is also applicable to wholly deterministic situations. Correlation moment analysis allows the analytic determination of the time dependence of the spatial moments of the solutions of certain types of non-linear partial differential equations. These moments provide measures of the growth of processes defined by the PDE, furthermore the results are obtained without requiring explicit solution of the PDE. The development is presented via case studies of the linear diffusion equation and the non-linear Kortweg de-Vries equation which indicate strategies for exploiting the various properties of correlation moments developed in the text. In addition, a variety of results have been developed which show how various classes of terms in PDEs affect the structure of a sequence of correlation moment equations. This allows results to be obtained about the behavior of the PDE solution, in particular how the presence of certain types of terms affects integral measures of the solution. It is also demonstrated that correlation moments provide a very simple, natural approach to determining certain subsets of conserved quantities associated with the PDEs.