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 and integrability of nonlinear equations that arise in soliton theory, dynamical systems, control theory, and mathematical physics. In particular, the software will gain insight in reaction-diffusion models, population and molecular dynamics, nonlinear networks, and chemical reactions. In addition to its use in research, the software will serve as an educational tool 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 operator methods 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 computer scientists, creating novel mathematics and quality software, as well as producing 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.
Hereman and his team created novel mathematical methods, algorithms, and symbolic software for the computation of conservation laws and Lax pairs of nonlinear models from the applied sciences and engineering. The existence of conserved quantities and a Lax pair guarantee that the mathematical equations model phenomena that obey physical conservation laws, such as preservation of mass, energy, momentum, or electrical charge. As an example, the ubiquitous Korteweg-de Vries equation has infinitely many conserved densities and a Lax pair which makes it a completely integrable nonlinear partial differential equation (PDE) that has been proven to accurately model water waves near beaches and ion-acoustic waves in plasmas. A major research activity was the development of software to compute conservation laws of nonlinear PDEs in multiple space dimensions, solely based on techniques from calculus, the calculus of variations, differential geometry, and linear algebra. It was therefore possible to implement the algorithms in Mathematica, a commonly used computer algebra language. Potential users are researchers concerned with conserved quantities and integrability of nonlinear PDEs that arise in soliton theory, dynamical systems, control theory, and mathematical physics. In particular, the software will help gain insight in reaction-diffusion models, population and molecular dynamics, nonlinear networks, and chemical reactions. An essential part of the project was the development of homotopy operator methods to handle mathematical integration and summation by parts. Fast algorithms and stand-alone packages were developed for continuous and discrete homotopy operators resulting in codes for use by non-experts who will no longer have to do the lengthy computations by hand. The software was applied to compute conservation laws of many nonlinear PDEs in two and three space dimensions. Doing so, Hereman's team was able to correct several errors in the literature. The research led to the Ph.D. Dissertation of one of the doctoral students involved in the project. Another major activity was the development of a code for the symbolic computation of Lax pairs of systems of nonlinear partial difference equations (lattices). Initially the team focused on scalar equations which are defined on a square but consistent around a cube. The resulting Mathematica program automatically computes Lax pairs for lattice versions of, e.g., the potential Korteweg-de Vries and sine-Gordon equations. Next, the approach was extended to nonlinear systems of lattices. The code can now handle various types of Boussinesq and Schroedinger lattices. The software was used to find new Lax pairs for novel systems of partial difference equations. The results will be presented in the Dissertation of the second Ph.D. student working on the project. Major findings came from the development of a scaling-based algorithm for the symbolic computation of Lax pairs for completely integrable PDEs. It turns out that a candidate Lax pair (L,M) can be constructed in three steps: select the order of the operator L and compute the order of M. Next, build a candidate Lax operator pair as linear combinations with undetermined coefficients of scaling invariant terms involving operators of lower order multiplied with the dependent variables and their derivatives. Finally, substitute the operators into the Lax equation and compute the undetermined coefficients. The algebraic systems to be solved are long and complicated and often Mathematica can neither reduce nor solve them. Groebner basis methods were crucial in handling such systems and considerable time was spent on finding optimal solution strategies. The new method is working well and Hereman's team has been able to construct Lax pairs for many nonlinear integrable PDEs. Here again, several mistakes were corrected in Lax pairs reported in the literature. The results were published in the Master of Science Theses of two graduate students. By the numbers, the project resulted in the publication of nine articles in peer-reviewed research journals, six chapters in books, three dissertations, and eight computer codes. Indeed, as a result of this project, comprehensive software packages are now available on the Internet to the research community. Hereman and his team envision that the software will have a wide impact with pure and applied mathematicians, physicists, engineers, and others interested in the analysis of nonlinear equations, thus enhancing the productivity of researchers working in those disciplines. In addition to its use in research, the software also serves as an educational tool for courses in nonlinear wave phenomena in fluid dynamics, plasma physics, electrical circuits, quantum chemistry, bio-genetics, and nonlinear optics. Hereman's research team was vertically integrated for it combined the strengths of seven undergraduate researchers, three graduate students, the principal investigator, and three visiting professors. Hereman presented the results of the project at numerous national and international meetings and promoted international collaboration, most notably, during a recent lecture tour in Turkey covering seven universities. This collaborative research project fostered interaction between mathematicians, physicists, and computer scientists. It led to novel mathematics and quality software and produced students with good software engineering skills.
