dNLS Flow on Discrete Space Curves

Abstract: The local induction equation, or the binormal flow on space curves is a well-known model of deformation of space curves as it describes the dynamics of vortex filaments, and the complex curvature is governed by the nonlinear Schr\"odinger equation (NLS). In this paper, we present its discrete analogue, namely, a model of deformation of discrete space curves by the discrete nonlinear Schr\"odinger equation (dNLS). We also present explicit formulas for both NLS and dNLS flows in terms of the $\tau$ function of the 2-component KP hierarchy.