Symbolic Software for the Study of Integrability of Nonlinear Partial Differential and Differential-Difference Equations

研究非线性偏微分和微分方程可积性的符号软件

• 批准号: 9901929
• 负责人: Willy Hereman
• 金额: $ 23.36万
• 依托单位: Colorado School of Mines
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-15 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9901929&HistoricalAwards=false
• 关键词: Symbolic; Software; Study; Integrability; Nonlinear

The objective of this project is to further develop and implement powerful symbolic algorithmsfor the investigation of integrability, through the calculation of conservation laws,higher-order symmetries, and recursion operators of nonlinear partial differential equations (PDES) and differential-difference equations (DDEs).Based on the concept of scaling invariance, the implemented algorithms will allow one to compute explicitly conserved densities, Hamiltonians, generalized symmetries, and cosymmetries. On the operator side, the algorithms can be used to compute the recursion operator and its conjugate, and the symplectic and cosymplectic (Hamiltonian) operators. The envisioned product is a comprehensive symbolic package in Mathematica syntax that will complement and build on previous software to compute invariants and symmetries for PDEs and DDEs.In addition, the project further develops symbolic software to compute exact solutions of PDEs and DDEs. The solution methods are based on multi-linear homogenization techniques. Once implemented, the software will allow the user to compute closed form solutions, such as soliton solutions and homoclinic and heteroclinic orbits, for large classes of nonlinear PDES. The extension of the method to DDEs will be investigated.The software will be written in Mathematica syntax. The packages will be entirely symbolic and produce analytical output. The development of novel mathematical techniques, in addition to the refinement and generalization of existing methods and algorithms, is an essential part of the proposed work. A comparison with existing algorithms will also be carried out.The aim is to provide high-quality symbolic software to researchers working on various aspects of integrability in soliton theory, dynamical systems, and mathematical physics. Specifically, our solvers and integrability testers are useful for the analysis of nonlinear PDEs and DDEs that arise in networks, elastic media, chemical kinetics, material sciences, biosciences, fluid dynamics, plasma physics, and nonlinear optics.

本项目的目标是通过计算非线性偏微分方程（PDES）和微分差分方程（DDE）的守恒律、高阶对称性和递归算子，进一步开发和实现用于可积性研究的强大符号算法。基于标度不变性的概念，所实现的算法将允许计算显式守恒密度、哈密顿量、广义对称和余对称。 在算子方面，该算法可用于计算递归算子及其共轭，以及辛和共辛（Hamilton）算子。 该产品是Mathematica语法的综合符号包，将补充和构建以前的软件，以计算偏微分方程和微分方程的不变量和对称性。此外，该项目还进一步开发符号软件，以计算偏微分方程和微分方程的精确解。 求解方法是基于多线性均匀化技术。 一旦实现，该软件将允许用户计算封闭形式的解决方案，如孤子解决方案和同宿和异宿轨道，为大类的非线性PDES。 本文将研究该方法在动态微分方程中的推广，并将用Mathematica语法编写软件。 这些资料包将完全是象征性的，并产生分析性的产出。 新的数学技术的发展，除了现有的方法和算法的细化和推广，是所提出的工作的一个重要组成部分。 与现有算法的比较也将进行。其目的是提供高质量的符号软件的研究人员在孤子理论，动力系统和数学物理的可积性的各个方面的工作。 具体而言，我们的求解器和可积性测试器可用于分析网络、弹性介质、化学动力学、材料科学、生物科学、流体动力学、等离子体物理和非线性光学中出现的非线性偏微分方程和微分方程。