The Darboux-Bäcklund transformation and Clifford algebras (Q2781678)

The author of this short paper considers a class of spectral problems in Clifford algebras. The main idea of the soliton surfaces approach consists in the following. Given a Zakharov-Shabat type spectral problem \(\Psi_{,x}=U\Psi \), \(\Psi_{,y}=V\Psi \), where \(U,V\) are matrices of a degree \(m\) and depending on \(x,y\) through some fields, \(\lambda \) is the spectral parameter and \(\Psi \) is a fundamental solution. Then the compatibility conditions read \(U_{,y}-V_{,x}+[U,V]=0\) and the known Sym-Tafel formula \(F=\Psi^{-1}\Psi_{\lambda }\), defines a \(\lambda\)-family of surfaces parametrized by \(x,y\). The surfaces are immersed in a space of matrices, which is a linear space. The Gauss-Codazzi equations for a class of surfaces given by \(F\) are equivalent to the considered integrable system. This approach is applied to produce some interesting classes of immersions associated with integrable systems. The known DB transformation is formulated in terms of Clifford numbers without using any matrix representation. The construction is a straightforward generalization of the DB transformation for isothermic surfaces.NEWLINENEWLINEFor the entire collection see [Zbl 0974.00040].