Symbolic Software for Conservation Laws of Multi-Dimensional Continuous and Discrete Nonlinear Equations

多维连续和离散非线性方程守恒律的符号软件Willy Hereman和Michael Colagros该项目的目标是创建数学方法、算法和符号软件，用于计算应用科学和工程中的非线性模型守恒律。具体地说，研究涉及多维非线性偏微分方程组、非线性微分差分方程组和完全离散的网格。使用变分中的微分几何工具和技术，算法直接在计算机代数语言中实现，该软件将易于非专家使用。潜在用户是研究孤子理论、动力系统、控制理论和数学物理中出现的非线性方程的守恒量和可积性的研究人员。特别是，该软件将深入了解反应扩散模型、种群和分子动力学、非线性网络和化学反应。除了在研究中使用外，该软件还将作为流体动力学、等离子体物理、电路、量子化学、生物遗传学和非线性光学中的非线性波动现象课程的教学工具。该项目的一个重要部分是开发同伦算子方法，以处理在喷射空间中所需的积分和部分求和。鉴于它们的广泛适用性，将为连续和离散同伦算子开发快速算法和独立程序包。遵循基于微积分的方法，将算法推广到积分-微分方程组和时滞微分方程组。新的数学技术有望从这些未知领域的探索中产生。这项研究促进了数学家和计算机科学家之间的合作，创造出新的数学和高质量的软件，并培养出具有良好软件工程技能的学生。软件开发过程，包括设计、开发和测试，结合了本科生、研究生和教师的优势。这些综合的软件包预计将对纯粹的和应用的数学家、物理学家、工程师和其他对非线性方程分析感兴趣的人产生广泛的影响，提高几个领域的研究人员的生产率。

Symbolic Software for Conservation Laws of Multi-Dimensional Continuous and Discrete Nonlinear EquationsWilly Hereman and Michael ColagrossoAbstract The aim of the project is to create mathematical methods, algorithms, and symbolic software for the computation of conservation laws of nonlinear models from the applied sciences and engineering.Specifically, the research concerns nonlinear partial differential equations in multi-dimensions, nonlinear differential-difference equations and fully-discretized lattices.Using differential-geometric tools and techniques from the calculus of variations, the algorithms are straightforward to implement in computer algebra languages and the software will be easy to use by non-experts. Potential users are researchers concerned with conserved quantities andintegrability of nonlinear equations that arise in soliton theory, dynamical systems, control theory, and mathematical physics. In particular, the software will gain insight in reaction-diffusionmodels, population and molecular dynamics, nonlinear networks, and chemical reactions.In addition to its use in research, the software will serve as an educationaltool for courses in nonlinear wave phenomena in fluid dynamics, plasma physics, electrical circuits, quantum chemistry, bio-genetics, and nonlinear optics.An essential part of the project is the development of homotopy operatormethods to handle the needed integration and summation by parts on jet spaces.In view of their versatile applicability, fast algorithms and stand-alone packages will be developed for continuous and discrete homotopy operators.Adhering to a calculus-based approach, the generalization of the algorithms to integro-differential equations and delay differential equations will be pursued. New mathematical techniques are expected to come from these explorations in uncharted terrain.The research fosters collaboration between mathematicians and computerscientists, creating novel mathematics and quality software, as well asproducing students with good software engineering skills. The software development process, including design, development, and testing,combines the strengths of undergraduates, graduate students, and faculty. The comprehensive software packages are envisioned to have a wide impact with pure and applied mathematicians, physicists, engineers, and others interested in the analysis of nonlinear equations, enhancing the productivity of researchers in several domains.