Connections of zero curvature and applications to nonlinear partial differential equations
Paul Bracken
Discrete & Continuous Dynamical Systems - S 2014, 7(6): 1165-1179 doi: 10.3934/dcdss.2014.7.1165
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 Bäcklund tranformations can be formulated for such equations that occur as zero curvature terms.
keywords: lax pair structure equations Connection Bäcklund. bundle curvature
Exterior differential systems and prolongations for three important nonlinear partial differential equations
Paul Bracken
Communications on Pure & Applied Analysis 2011, 10(5): 1345-1360 doi: 10.3934/cpaa.2011.10.1345
Partial differential systems which have applications to water waves will be formulated as exterior differential systems. A prolongation structure is determined for each of the equations. The formalism for studying prolongations is reviewed and the prolongation equations are solved for each equation. One of these differential systems includes the Camassa-Holm and Degasperis-Procesi equations as special cases. The formulation of conservation laws for each of the systems introduced is discussed and a single example for each is given. It is shown how a Bäcklund transformation for the last case can be obtained using the prolongation results.
keywords: prolongation connection integral manifolds. Exterior differential system

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