Communications on Pure and Applied Analysis (CPAA)

Exterior differential systems and prolongations for three important nonlinear partial differential equations

Pages: 1345 - 1360, Volume 10, Issue 5, September 2011      doi:10.3934/cpaa.2011.10.1345

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Paul Bracken - Department of Mathematics, University of Texas, Edinburg, TX 78539, United States (email)

Abstract: 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 B├Ącklund transformation for the last case can be obtained using the prolongation results.

Keywords:  Exterior differential system, prolongation, connection, integral manifolds.
Mathematics Subject Classification:  Primary: 35G20, 35Q53, 37J35; Secondary: 37K25, 37K35, 55R10, 53Z05.

Received: February 2009;      Revised: August 2010;      Available Online: April 2011.