Connections of zero curvature and applications to nonlinear partial differential equations

Paul Bracken

doi:10.3934/dcdss.2014.7.1165

Article Contents

2014, Volume 7, Issue 6: 1165-1179. Doi: 10.3934/dcdss.2014.7.1165

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Connections of zero curvature and applications to nonlinear partial differential equations

Paul Bracken^1,

1.
Department of Mathematics, University of Texas, Edinburg, TX, 78540

Received: December 31, 2012

Revised: August 31, 2013

Published: November 2014

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Abstract

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.

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Mathematics Subject Classification: Primary: 53Z05, 57R99; Secondary: 55R10, 53B50.

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