Symbolic Software for the Investigation of Nonlinear Partial Differential Equations

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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. ***