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

Abstract

A framework which combines Green’s function (GF) methods and techniques from the theory of stochastic processes is proposed for tackling nonlinear evolution problems. The framework, established by a series of easy-to-derive equivalences between Green’s function and stochastic representative solutions of linear drift–diffusion problems, provides a flexible structure within which nonlinear evolution problems can be analyzed and physically probed. As a preliminary test bed, two canonical, nonlinear evolution problems – Burgers’ equation and the nonlinear Schrödinger’s equation – are first treated. In the first case, the framework provides a rigorous, probabilistic derivation of the well known Cole–Hopf ansatz. Likewise, in the second, the machinery allows systematic recovery of a known soliton solution. The framework is then applied to a fairly extensive exploration of physical features underlying evolution of randomly stretched and advected Burger’s vortex sheets. Here, the governing vorticity equation corresponds to the Fokker–Planck equation of an Ornstein–Uhlenbeck process, a correspondence that motivates an investigation of sub-sheet vorticity evolution and organization. Under the assumption that weak hydrodynamic fluctuations organize disordered, near-molecular-scale, sub-sheet vorticity, it is shown that these modes consist of two weakly damped counter-propagating cross-sheet acoustic modes, a diffusive cross-sheet shear mode, and a diffusive cross-sheet entropy mode. Once a consistent picture of in-sheet vorticity evolution is established, a number of analytical results, describing the motion and spread of single, multiple, and continuous sets of Burger’s vortex sheets, evolving within deterministic and random strain rate fields, under both viscous and inviscid conditions, are obtained. In order to promote application to other nonlinear problems, a tutorial development of the framework is presented. Likewise, time-incremental solution approaches and construction of approximate, though otherwise difficult-to-obtain backward-time GF’s (useful in solution of forward-time evolution problems) are discussed.