Applications of Symmetry Methods in Continuum Mechanics

许多物理现象都是由微分方程式模拟的，这些微分方程式将感兴趣的物理参数及其变化率联系在一起。当代的大多数模型都是非线性的，也就是说，控制方程的两个解之和不是一个解。非线性现象的一个简单例子是相互作用的水波，其中产生的波的高度不等于每个波的高程之和。非线性效应使得非线性微分方程组的精确通解几乎是不可能的。对于某些类型的方程已经开发了特定的求解技术，但这些技术缺乏通用性；数值解缺乏精度和灵活性，特别是在三维情况下，必须使用超大的数据结构来实现足够的精度。当前建议中所考虑的微分方程对称理论适用于广泛的非线性模型；众所周知，它能够产生有用的精确解和关于潜在现象的分析信息。对称是变换变量但保留描述模型的方程的任何变换。在对称变换的作用下，方程被简单地映射成它们自己。可以系统地计算微分方程组的局部对称。当已知时，它们用于某些模型(由常微分方程给出)的降阶和完全解，以及其它模型(由偏微分方程给出)的自相似和其他类型的精确解。目前的建议包括理论和应用部分。第一个理论部分致力于不具有或具有太少局部对称性的微分方程的对称性的推导和研究。这里采用的方法是寻找增广(势)方程组的额外对称性，其中包含额外的非局部(势)变量。这种方法最近对一些感兴趣的应用产生了新的结果；它旨在进一步推广和扩展非局部对称框架。第二个理论部分是关于小参数和/或不同时空尺度的方程的近似对称性，即近似成立的对称性。到目前为止，还没有关于近似对称的统一理论。在拟议的研究项目中，计划对对称性理论进行一致的扩展，以包括近似对称性，并研究构建非线性模型近似解的系统方法。所提出的研究的应用部分在于寻求由连续介质力学中的非线性问题的动力学守恒的对称性、精确解和量，包括流体、气体、等离子体和非线性弹性介质的动力学。特别是，我们计划与两组合作者一起研究弹性固体中的波传播模型，以及与不可压缩流体中的湍流模型有关的方程。还计划对申请人编写的符号对称计算软件包GEM for Maple进行扩展和改进。该包目前正被全球相当数量的研究人员使用。

Many physical phenomena are modeled by differential equations, which relate the physical parameters of interest and their rates of change. The majority of contemporary models are nonlinear, that is, a sum of two solutions to the governing equations is not a solution. A simple example of a nonlinear phenomenon is provided by interacting water waves, where the height of the resulting wave is not equal to the sum of elevations of each of the waves. Nonlinear effects make the exact general solution of nonlinear differential equations virtually impossible. Specific solution techniques have been developed for some classes of equations, but such techniques lack generality; numerical solutions lack precision and flexibility, especially in three-dimensions, where extra large data structures have to be used to achieve adequate precision.The theory of symmetries of differential equations considered in the current proposal is applicable to wide classes of nonlinear models; it is known to be able to yield useful exact solutions and analytical information about the underlying phenomenon. A symmetry is any transformation that transforms variables, but preserves the equations describing the model. Under the action of a symmetry transformation, the equations are simply mapped into themselves. Groups of local symmetries of differential equations can be systematically computed. When known, they are used for the reduction of order and complete solution for some classes of models (given by ordinary differential equations), and for obtaining self-similar and other types of exact solutions of other models (given by partial differential equations).The current proposal consists of theoretical and applied parts.The first theoretical part is devoted to the derivation and study of symmetries of differential equations which do not have, or have too few, local symmetries. The approach taken here is to seek extra symmetries of augmented (potential) systems of equations, which contain additional nonlocal (potential) variables. This approach has recently yielded new results for some applications of interest; it is intended to further generalize and extend the nonlocal symmetry framework. The second theoretical part is devoted to the approximate symmetries, that is, symmetries that hold approximately, for equations that involve small parameters and/or different time/space scales. To date, there is no unified theory of approximate symmetries. Within the proposed research project, it is planned to work on a consistent extension of the symmetry theory to include approximate symmetries, and on systematic methods of construction of approximate solutions for nonlinear models. The applied part of the proposed research consists in seeking symmetries, exact solutions, and quantities conserved by the dynamics of nonlinear problems in continuum mechanics, including dynamics of fluids, gases, plasmas, and nonlinear elastic media. In particular, with two groups of collaborators, we plan to study models of wave propagation in elastic solids, and equations pertaining to turbulence modelling in incompressible fluids. It is also planned to work on the extension and improvement of the symbolic symmetry computation software package GeM for Maple, written by the applicant. The package is currently used by a significant number of researchers around the globe.