Marvan M. Classification of integrable PDE in the differential geometry of surfaces, talk at Conf. The Interface of Integrability and Quantization, Lorentz Center, Leiden (The Netherlands), 2010 (abstract)
Speaker: Michal Marvan
Title: Classification of integrable PDE in the differential geometry of surfaces
The purpose of this talk is to present a recent project, undertaken with my collaborators from the Silesian university in Opava, and the results obtained so far.
Recognizing integrability is among the important unsolved problems in soliton theory. Numerous beautiful results have been obtained by indirect approaches like the singularity analysis and the symmetry analysis. Yet obtaining a reasonably complete classification of integrable PDE remains a challenge. In particular, the majority of classification problems in differential geometry seem to be beyond the scope of the methods mentioned.
To cope with the problem, we apply the method of characteristic elements dating back to 1992.
Classifying equations possessing a zero curvature representation with values in a prescribed unsolvable Lie algebra is equivalent to solving a quasilinear system of equations in total derivatives. However, unless in simplest settings, the calculations are prohibitively resource-demanding. Moreover, one-parametric families of zero curvature representations, which are characteristic of integrability, have to be selected from the vast corpus of calculation results.
Slides: Marvan M. Classification of integrable PDE in the differential geometry of surfaces (presentation at the Lorentz Center Workshop The Interface of Integrability and Quantization, Leiden, The Netherlands, 12-16 April 2010).pdf