Horizontal gauge cohomology of PDE's

Zero-curvature representations (ZCR) with values in a non-solvable Lie algebra and their 1-parametric families are important attributes of completely integrable nonlinear systems of PDE's, especially in dimension two.

While conservation laws and horizontal cohomology in general are computable via the so called C-spectral sequence (A.M. Vinogradov, 1978), for ZCR's the corresponding cohomology (gauge cohomology) was introduced only in 1993. Gauge cohomology is related to ZCR's in much the same way as horizontal cohomology is related to conservation laws (the latter reduces to the former when the Lie algebra is Abelian).

In particular, a procedure of computing zero-curvature representations arises, which appears to be, unlike the now classical Wahlquist--Estabrook procedure, applicable to classification problems even in absence of limits on the nature of ZCR's other than irreducibility. Moreover, horizontal gauge cohomology provides obstructions to removability of the spectral parameter of the ZCR.