On generalized Bäcklund transformations for equations describing pseudo-spherical surfaces

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.