The Bäcklund transformations, exact solutions, and conservation laws for the compound modified Korteweg-de Vries-Sine-Gordon equations which describe pseudospherical surfaces

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. © 2013 S. M. Sayed.