Abstract: In this talk, I will describe a new, transformation group theoretic, approach to the classification of Darboux integrable partial differential equations of the type
uxy = f(x, y, u, ux, uy).
Such equations are commonly referred to in the literature as f-Gordon equations, generalized wave map equations, or equations of Liouville type.
The main result of this group theoretic approach is that a complete list of all f-Gordon equations which are Darboux integrable at order 3 can be determined from a complete list of rank 2 distributions in 5 dimensions which admit intransitive 5-dimensional symmetry groups. In this way, the study of Darboux integrable f-Gordon equations is tied, quite remarkably, to Cartan's celebrated 1910 paper. Through this correspondence, we uncovered a new class of Darboux integrable equations which leads to a complete classification of all Darboux integrable equations at order 3.
This talk is based on joint work with Brandon P. Ashley, Southern Oregon University.