Connections of Zero Curvature and Applications to Nonlinear Partial Differential Equations

A general formulation of zero curvature connections in a principle bundle is presented and some applications are discussed. It is proved that a related connection based on a prolongation in an associated bundle remains zero curvature as well. It is also shown that the connection coefficients can be defined so that the partial differential equation to be studied appears as the curvature term in the structure equations. It is discussed how Lax pairs and Backlund transformations can be formulated for such equations. It is discussed how Lax pairs and Backlund transformations can be formulated for such equations that occur as zero curvature terms.