On the geometry of pseudopotentials of evolution equations (Q1909838)

The article is devoted to the geometric aspects of the theory of evolution differential equations related to the notion of a pseudopotential which was introduced by Wahlquist and Estabrook in 1975. In [J. Math. Phys. 16, No. 1, 1-7 (1975; Zbl 0298.35012)], \textit{H. D. Wahlquist} and \textit{F. B. Estabrook} have constructed a geometrical theory of pseudopotentials of evolution equations of the second order on the base of Cartan-Laptev invariant method. In a similar manner one can construct a theory of pseudopotentials of equations of arbitrary order. During the investigation, we had to analyze and refine the proper notion of a pseudopotential (we should also note that some authors impose certainly artificial restrictions upon pseudopotentials introduced by them). We prove possibility of constructing the representations of zero curvature for an arbitrary evolution equation of the second order with one space variable. This, in particular, implies the existence of an infinite set of conservation laws for an arbitrary evolution equation of the second order.