On Complete Integrability of the Generalized Weierstrass System

Abstract

In this paper we study certain aspects of the complete integrability of the Generalized Weierstrass system in the context of the Sinh-Gordon type equation. Using the conditional symmetry approach, we construct the Bäcklund transformation for the Generalized Weierstrass system which is determined by coupled Riccati equations. Next a linear spectral problem is found which is determined by nonsingular 2 × 2 matrices based on an sl(2, C) representation. We derive the explicit form of the Darboux transformation for the Weierstrass system. New classes of multisoliton solutions of the Generalized Weierstrass system are obtained through the use of the Bäcklund transformation and some physical applications of these results in the area of classical string theory are presented.

The Generalized Weierstrass System for Nonconstant Mean Curvature Surfaces and the Nonlinear Sigma Model

“…In particular, many properties of such surfaces determined by Konopelchenko's inducing prescription, such as the relationship to the nonlinear two-dimensional sigma model, integrability and Lax pair have been determined and a Bäcklund transformation has been calculated as well [7,8].…”