Seminar talk, 13 December 2011

Speaker: Michal Marvan

Title: Integrable surfaces

Abstract:
It is an open problem to decide whether a given nonlinear partial differential equation (PDE) is integrable in the sense of soliton theory. It is an even more challenging problem to classify all integrable systems within a given class of PDEs. Possible ways of tackling these problems can be direct and indirect.

The indirect methods, which now dominate the field, stem from plausible but yet unproven conjectures, such as the conjecture that in the case of generic evolution systems, sufficiently many higher symmetries imply integrability. The main problem with the indirect methods is that they cannot give reliable and exhaustive answers until the underlying conjecture and its converse are proved.

The direct methods look for the structure that actually underlies an integration method, e.g., for a zero curvature representation or a Lax pair. The direct methods have a potential to give reasonably exhaustive answers (modulo technical restrictions). Moreover, only the direct methods can validate the indirect methods. Perhaps the best known of the direct methods is the Wahlquist-Estabrook method and its substantial developments. However, as of now none of these methods is suitable for solving serious classification problems.

Computation of characteristic elements underlies another direct method, tailored to an unknown PDEs and a given Lie algebra the zero curvature representation takes values in. The problem is reduced to solving a system of differential equations in total derivatives of some modest degree of nonlinearity. The system becomes linear if a seed zero curvature representation without parameter is known. This is the case, e.g., with immersed surfaces in the Euclidean space. It was in this area where recent classification efforts brought new results that have not been obtained by indirect methods even though such attempts have been performed.