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A framework which combines Green's function (GF) methods and techniques fromthe theory of stochastic processes is proposed for tackling nonlinear evolutionproblems. The framework, established by a series of easy-to-derive equivalencesbetween Green's function and stochastic representative solutions of lineardrift-diffusion problems, provides a flexible structure within which nonlinearevolution problems can be analyzed and physically probed. As a preliminary testbed, two canonical, nonlinear evolution problems - Burgers' equation and thenonlinear Schrodinger's equation - are first treated. In the first case, theframework provides a rigorous, probabilistic derivation of the well knownCole-Hopf ansatz. Likewise, in the second, the machinery allows systematicrecovery of a known soliton solution. The framework is then applied to a fairlyextensive exploration of physical features underlying evolution of randomlystretched and advected Burger's vortex sheets. Here, the governing vorticityequation corresponds to the Fokker-Planck equation of an Ornstein-Uhlenbeckprocess, a correspondence that motivates an investigation of sub-sheetvorticity evolution and organization. Under the assumption that weakhydrodynamic fluctuations organize disordered, near-molecular-scale, sub-sheetvorticity, it is shown that these modes consist of two weakly dampedcounter-propagating cross-sheet acoustic modes, a diffusive cross-sheet shearmode, and a diffusive cross-sheet entropy mode. Once a consistent picture ofin-sheet vorticity evolution is established, a number of analytical results,describing the motion and spread of single, multiple, and continuous sets ofBurger's vortex sheets, evolving within deterministic and random strain ratefields, under both viscous and inviscid conditions, are obtained.

annals of physics

Green's function-stochastic methods framework for probing nonlinear  evolution problems: Burger's equation, the nonlinear Schrodinger's equation,  and hydrodynamic organization of near-molecular-scale vorticity