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 B\"{a}cklund 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 $2 \times 2$ matrices based on an $sl (2, \mathbb C)$ 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 B\"{a}cklund transformation and some physical applications of these results in the area of classical string theory are presented.