Applied Mathematics Seminar

Riemann's method is one of the definitive ways of finding the fundamental solution for a linear hyperbolic PDE in two variables. The idea of applying Lie Point Symmetries to finding Riemann functions is well established.

In this talk we shall review these results and show how they are in fact nearly all isomorphic to Riemann's original example, the EPD equation. At the same time though, Riemann functions have been found which do not admit this equivalence. Nevertheless we shall show that if one looks beyond point symmetries to the logical extension of Lie-Backlund symmetries then these results are in fact transformable to sub-cases of the most general self-adjoint equation for which the Riemann function is known, namely Chaundy's equation. By seeking the generalised symmetries of this equation, a new equivalence class of Riemann functions obtainable only by a Lie-Backlund symmetry is derived. In conclusion we shall connect these results to the inverse problem of scattering theory.