Symmetry methods and their applications to the analysis of modern mathematical models

物理现象的数学模型通常用偏微分方程（PDE）来表示，将物理参数与其在时间和空间中的变化率联系起来。PDE模型出现在各种情况下，包括物理连续体的描述（例如，流体、等离子体、混合物、气体和固体）。大多数复杂系统的数学模型是非线性的：控制方程的两个解的和不是解。非线性现象的例子包括水波、弹性固体的大变形和湍流流动。提出的研究是关于数学方法的发展，可以系统地应用于分析，简化，并解决非线性偏微分方程。对于PDE模型，理想情况下希望有一个根据显式公式的解决方案。这将允许以直接的方式进行计算、绘制图形和分析解的行为。对于线性模型，存在许多经典的求解方法，但对于非线性偏微分方程，通常的方法不适用；研究人员经常必须提出特殊的方法，或者使用数值模拟，这可能很耗时，而且可能无法提供“大局”。本提案的理论部分致力于偏微分方程对称性理论的发展和扩展。适用于广泛的线性和非线性模型，对称方法用于系统地提取有关问题的重要信息，简化问题，构造精确解，在某些情况下，完全解决给定的非线性问题。然而，对于一些感兴趣的偏微分方程系统，经典对称方法产生的有用结果很少。在这些情况下，对称方法的扩展证明是有益的。申请人将利用在局部/非局部对称理论、精确解和符号计算方面的专业知识和研究势头，以及加拿大和国际合作和HQP，进一步发展与系统对称相关的方法，特别是非局部和近似对称理论，以分析和解决以前棘手的非线性问题。该建议的第二部分与复杂PDE系统的对称性相关计算问题有关，其中需要计算机辅助计算。申请人是基于maple的GeM符号包的作者，该包被许多国家的研究人员使用。在即将到来的授权周期中，它打算使GeM面向对象，允许更高的效率和更简单的语法，并且还实现额外的例程来计算对称性相关的数学对象，包括线性化算子，伴随和近似对称性，以及拉格朗日量。当前的建议包括一个应用部分。申请人将与合作者一起研究波在各向异性弹性固体中的非线性传播模型和湍流模型，并使用先进的对称方法研究相应的偏微分方程。

Mathematical models of physical phenomena are often stated in terms of partial differential equations (PDE), relating physical parameters to their rates of change in time and space. PDE models arise in a variety of contexts, including the description of physical continua (e.g., fluids, plasmas, mixtures, gases, and solid bodies). The majority of mathematical models of complex systems are nonlinear: a sum of two solutions of governing equations is not a solution. Examples of nonlinear phenomena include water waves, large deformations of elastic solids, and turbulent fluid flows. The proposed research is concerned with the development of mathematical methods that can be systematically applied to analyze, simplify, and solve nonlinear PDEs. For a PDE model, one ideally would like to have a solution in terms of explicit formulas. This would allow to make computations, plot graphs, and analyze solution behaviour in a direct way. For linear models, many classical solution methods exist, but for nonlinear PDEs, usual methods are not applicable; researchers often must come up with special approaches, or use numerical simulations, which can be time consuming, and may not provide a "big picture". The theoretical part of this proposal is devoted to the development and extension of the theory of symmetries of PDEs. Applicable to wide classes of linear and nonlinear models, symmetry methods are used to systematically extract important information about the problem, simplify it, construct exact solutions, and in some cases, completely solve a given nonlinear problem. For some PDE systems of interest, however, classical symmetry methods yield few useful results. In these cases, extensions of symmetry methods prove beneficial. The applicant will capitalize on expertise and research momentum in theory of local/nonlocal symmetries, exact solutions, and symbolic computations, as well as on Canadian and international collaborations and HQP, to develop further systematic symmetry-related methods, in particular, the theory of nonlocal and approximate symmetries, to analyze and solve previously intractable nonlinear problems. The second part of this proposal has to do with the problem of symmetry-related computations for complex PDE systems, where computer-aided calculations are required. The applicant is the author of a Maple-based symbolic package GeM, used by researchers in many countries. In the upcoming grant cycle, it is intended to make GeM object-oriented, allowing for higher efficiency and simpler syntax, and also to implement additional routines to compute symmetry-related mathematical objects, including linearlization operators, adjoint and approximate symmetries, and Lagrangians. The current proposal includes an applied part. With collaborators, the applicant will work on nonlinear models of wave propagation in anisotropic elastic solids, and models pertaining to turbulent flows, and study the corresponding PDEs using advanced symmetry methods.