The Theory of Connections and the Problem of Existence of Backlund Transformations for Second Order Evolution Equations

Abstract

Backlund transformations are used to search for solutions, particularly soliton solutions, of non-linear differential equations. In this paper we present an invariant geometrical theory of Backlund transformations for second order evolution equations with one space variable. The main concept is that of connection defining the representation of zero curvature for a given partial differential equation. The main result of this paper is a criterion of existence of Backlund transformations for second order evolution equations with one space variable. We find thegeneral form of a second order evolution equation that admits Backlund transformations. Furthermore, for a special important class of evolution equations we show that a Backlund transformation exists if and only if the equation has one of two special forms. An equation of the first of these types can be then reduced, by a change of variable, to the Burgers equation, and the equation of the second type can be reduced to a well-known linear equation. All differential-geometric considerations in this paper are local.