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

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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.