Applications of Symmetry Methods in Continuum Mechanics

许多物理现象都是由微分方程来模拟的，微分方程将感兴趣的物理参数及其变化率联系起来。大多数当代模型是非线性的，也就是说，两个解的总和的控制方程是不是一个解决方案。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度之和。 非线性效应使得非线性微分方程的精确通解实际上是不可能的。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，特别是在三维中，必须使用超大的数据结构来实现足够的精度。 在目前的建议中考虑的微分方程的对称性理论适用于广泛的非线性模型类，它是已知的，能够产生有用的精确解和分析信息的基本现象。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，方程被简单地映射到自身。微分方程的局部对称群可以系统地计算。当已知时，它们用于某些类型的模型（由常微分方程给出）的降阶和完全解，以及用于获得其他模型（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。 第一理论部分致力于推导和研究微分方程的对称性，这些微分方程没有或很少有局部对称性。这里采用的方法是寻求增广（势）方程组的额外对称性，其中包含额外的非局部（势）变量。这种方法最近产生了一些感兴趣的应用新的结果，它的目的是进一步推广和扩展的非局部对称性框架。 第二个理论部分是致力于近似对称性，即近似保持的对称性，涉及小参数和/或不同的时间/空间尺度的方程。到目前为止，还没有统一的近似对称理论。在拟议的研究项目中，计划研究对称理论的一致扩展，以包括近似对称，并研究构建非线性模型近似解的系统方法。 拟议的研究的应用部分包括在寻求对称性，精确解，并在连续介质力学，包括流体，气体，等离子体和非线性弹性介质的动力学的非线性问题的动力学守恒量。特别是，与两组合作者，我们计划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建模有关的方程。 还计划扩展和改进申请人编写的Maple符号对称计算软件包GeM。该软件包目前被地球仪的大量研究人员使用。

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.