Various dynamical systems are described by differential equations. In control theoretic problems (e.g., in robotics), it is essential to determine and utilise the degrees of freedom of a system. The aim of this project is to develop new algorithmic methods for determining integrability conditions for systems of partial differential equations (PDEs). Certain effective techniques in differential algebra are readily available, e.g., to determine all power series solutions of a system of PDEs. However, for concepts such as Bäcklund transformations or Lax pairs, which play a significant role in the theory of integrable systems, no systematic effective way of finding these relationships between systems of PDEs is known. This project will build on differential geometry, jet calculus, Lie symmetries and differential algebra to approach these concepts.
The new methods should be implemented in computer algebra software, preferably as an extension of existing Maple packages, such as Janet and DifferentialThomas. The available resources for High Performance Computing should be used, developing parallelised methods whenever possible.
Second Supervisor: Dr Colin Christopher