Theor. and Math. Physics (russian)

Minimal surfaces in multidimensional spaces naturally arise as classical trajectories of relativistic strings with Lagrangians suggested by Nambu (see [1]) and by Polyakov (see [2]). From geometric point of view these are surfaces of zero mean curvature. Their immersion into environment space often is given by the equations integrable by means of inverse scattering method (see [3–5]). Minimal surfaces in R and R are described by Liouville equation uzz̄ = e . This equation is nonlinear, but can be linearized by Backlund transformation (see [6, 7]). In higher dimensional spaces and in non-flat spaces immersion equations for minimal surfaces are not linearizable. But they possess Lax pairs (see [5]), therefore one can rather effectively study their solutions. In this paper minimal tori in five-dimensional sphere S ⊂ C 3 are considered whose immersion is described by Bullough-Dodd-Jiber-Shabat equation uzz̄ = e u − e. (1.1)