The Bäcklund problem for the equation ∂2z/∂x1∂x2= f (z)

Abstract

The Bäcklund problem for the equation ∂²z/∂x¹∂x²=f (z) is discussed for analytic functions f, using the procedure of Estabrook and Wahlquist, starting from a Lagrangian formulation. The condition d²f/dz² =kf, k constant, necessary for the existence of nontrivial Bäcklund maps when the space of new dependent variables is R, is shown to be closely related to the structure of the Lie algebra SL(2,R).