On generalized Bäcklund transformations for equations describing pseudo-spherical surfaces

Abstract

It is shown that each one-parameter subgroup of SL(2, R ) gives rise to a local correspondence theorem between suitably generic solutions of arbitrary scalar equations describing pseudo-spherical surfaces. Thus, if appropriate genericity conditions are satisfied, there exist local transformations between any two solutions of scalar equations arising as integrability conditions of sl(2, R ) -valued linear problems. A complete characterization of evolution equations u t = K( x, t, u, u x ,…, u x k ) which are of strictly pseudo-spherical type is also provided.