The Darboux-Bianchi-Bäcklund transformation and soliton surfaces (1303.5472v1)

Abstract: In the first part of the paper we present the dressing method which generates multi-soliton solutions to integrable systems of nonlinear partial differential equations. We compare the approach of Neugebauer with that of Zakharov, Shabat and Mikhailov. In both cases we discuss the group reductions and reductions defined by some multilinear constraints on matrices of the linear problem. The second part of the paper describes the soliton surfaces approach. The so called Sym-Tafel formula simplifies the explicit reconstruction of the surface from the knowledge of its fundamental forms, unifies various integrable nonlinearities and enables one to apply powerful methods of the theory of solitons to geometrical problems. The Darboux-Bianchi-B\"acklund transformation (i.e., the dressing method on the level of soliton surfaces) reconstructs explicitly many classical transformations of XIX century. We present examples of interesting classes of surfaces obtained from spectral problems. In particular, we consider spectral problems in Clifford algebras associated with orthogonal coordinates. Finally, compact formulas for multi-soliton surfaces are discussed and applied for the Localized Induction Equation with axial flow.