Prolongation structures of nonlinear evolution equations

Abstract

A technique is developed for systematically deriving a ’’prolongation structure’’—a set of interrelated potentials and pseudopotentials—for nonlinear partial differential equations in two independent variables. When this is applied to the Korteweg−de Vries equation, a new infinite set of conserved quantities is obtained. Known solution techniques are shown to result from the discovery of such a structure: related partial differential equations for the potential functions, linear ’’inverse scattering’’ equations for auxiliary functions, Bäcklund transformations. Generalizations of these techniques will result from the use of irreducible matrix representations of the prolongation structure.