Symbolic Software for the Investigation of Nonlinear Partial Differential Equations

这个项目继续开发和实现新的强大的符号算法，用于测试可积性和守恒定律的计算。还将研究非线性偏微分方程组(PDE)闭式解的构造。特别是，对于非线性发展方程组，第一个符号软件包将自动生成多项式型守恒律。基于方程的标度性质(对称性)，该算法提供了一种构造守恒密度及其相关通量的巧妙方法。基于齐次化技术，第二个软件包将允许用户计算大类偏微分方程组的特定解的解析形式。对于可积方程，该方法可得到孤子解。对于不可积方程，用该方法可以有效地计算同宿轨和异宿轨的闭式表达式。即将开发的符号软件将补充和建立早期的算法，用于计算李点和非经典对称性，通过广田直接法寻找孤子解，并通过Painleve检验来测试微分方程组的可积性。这些程序将用Macsyma和MATHEMICA语法编写。它们将完全是象征性的，并产生分析结果。除了改进和推广现有的方法和算法外，开发新的数学技巧也是这项研究项目的一部分。其目的是为从事孤子理论、动力学系统、数学物理，特别是网络中的非线性波动现象、弹性介质、化学动力学、生物科学、流体动力学、等离子体物理和非线性光学的研究人员提供高质量的符号包。***

This project continues the development and implementation of new powerful symbolic algorithms for testing the integrability and the calculation of conservation laws. The construction of closed- form solutions for systems of nonlinear partial differential equations (PDEs) will also be investigated. In particular, for systems of nonlinear evolution equations, a first symbolic software package will automatically generate polynomial-type conservation laws. Based on scaling properties (symmetries) of the equations, the proposed algorithm provides an elegant way to construct conserved densities and their associated fluxes. Based on homogenization techniques, a second software package will allow the user to compute the analytic form of particular solutions for large classes of PDEs. For integrable equations, the technique leads to soliton solutions. For non-integrable equations, closed-form expressions of homoclinic and heteroclinic orbits can be computed efficiently with the method. The symbolic software to be developed will complement and build on earlier algorithms for the computation of Lie-point and non- classical symmetries, to find soliton solutions via Hirota's direct method, and to test the integrability of differential equations via the Painleve test. The programs will be written in Macsyma and Mathematica syntax. They 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 also a part of this research project. The aim is to provide high-quality symbolic packages to researchers working on soliton theory, dynamical systems, mathematical physics; specifically nonlinear wave phenomena in networks, elastic media, chemical kinetics, bio-sciences, fluid dynamics, plasma physics, and nonlinear optics. ***