Applications of Symmetry Methods in Continuum Mechanics

许多物理现象都是用微分方程来模拟的，微分方程与感兴趣的物理参数及其变化率有关。大多数现代模型是非线性的，也就是说，控制方程的两个解的和不是解。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度之和。非线性效应使得非线性微分方程的精确通解几乎不可能。对于某些类型的方程，已经开发出了特定的解技术，但这种技术缺乏通用性；数值解缺乏精度和灵活性，特别是在三维空间中，必须使用超大的数据结构才能达到足够的精度。**当前建议中考虑的微分方程的对称性理论适用于各种非线性模型；众所周知，它能够产生关于潜在现象的有用的精确解和分析信息。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，方程被简单地映射到自身。微分方程的局部对称性群可以系统地计算。当已知时，它们用于某些模型（由常微分方程给出）的阶约和完全解，以及获得其他模型（由偏微分方程给出）的自相似和其他类型的精确解。**目前的建议由理论和应用两部分组成。**第一个理论部分致力于不具有局部对称性或局部对称性过少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包含额外的非局部（势）变量的增广（势）方程组的额外对称性。这种方法最近在一些感兴趣的应用中产生了新的结果；目的是进一步推广和扩展非局部对称框架。**理论的第二部分是关于近似对称性的，也就是说，对于涉及小参数和/或不同时间/空间尺度的方程来说，近似对称性是成立的。到目前为止，还没有关于近似对称性的统一理论。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，以及非线性模型近似解的系统构建方法。**本研究的应用部分在于寻求连续介质力学中非线性问题动力学的对称性、精确解和守恒量，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，以及不可压缩流体中湍流模型的方程。**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。该软件包目前被全球大量研究人员使用。

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.