探索非线性演化问题的格林函数-随机方法框架：Burger方程，非线性薛定谔方程和近分子尺度涡度的流体动力学组织
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
摘要：
提出了一种结合格林函数（GF）方法和随机过程理论的技术来处理非线性演化问题的框架。该框架，建立了一系列容易推导的格林函数和随机代表性的线性漂移扩散问题的解决方案之间的等价关系，提供了一个灵活的结构，其中非线性演化问题可以分析和物理探测。作为一个初步的测试床，两个典型的，非线性发展问题- Burgers方程和非线性薛定谔方程-首先处理。在第一种情况下，该框架提供了一个严格的，概率推导的著名的科尔-霍普夫的anastomy。同样，在第二，机器允许系统恢复一个已知的孤子解决方案。该框架，然后应用到一个相当广泛的探索随机拉伸和平流汉堡的涡面演变的物理特征。在这里，控制涡度方程对应于福克-普朗克方程的奥恩斯坦-乌伦贝克过程，一个对应关系，激发调查的子片涡度的演变和组织。在弱流体动力学波动组织无序的、近分子尺度的子片涡度的假设下，研究表明这些模式由两个弱阻尼反向传播的跨片声学模式、扩散跨片剪切模式和扩散跨片剪切模式组成。表熵模式。一旦建立了一致的画面，在表涡度演变，一些分析结果，描述的运动和传播的单个，多个，和连续的套伯格涡面，确定性和随机应变率场内的发展，粘性和无粘性条件下，得到。
英文摘要：
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 Schrodinger'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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
