Journal of Applied Mathematics
Volume 2013 (2013), Article ID 613065, 7 pages
http://dx.doi.org/10.1155/2013/613065
Research Article

The Bäcklund Transformations, Exact Solutions, and Conservation Laws for the Compound Modified Korteweg-de Vries-Sine-Gordon Equations which Describe Pseudospherical Surfaces

S. M. Sayed1,2

1Mathematics Department, Faculty of Science, Tabouk University, Tabouk, Saudi Arabia
2Mathematics Department, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

Received 14 November 2012; Revised 21 January 2013; Accepted 24 January 2013

Academic Editor: Laurent Gosse

Copyright © 2013 S. M. Sayed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

I show that the compound modified Korteweg-de Vries-Sine-Gordon equations describe pseudospherical surfaces, that is, these equations are the integrability conditions for the structural equations of such surfaces. I obtain the self-Bäcklund transformations for these equations by a geometrical method and apply the Bäcklund transformations to these solutions and generate new traveling wave solutions. Conservation laws for the latter ones are obtained using a geometrical property of these pseudospherical surfaces.