On Two Types of Localized Induction Equation

Abstract

We are looking for motions of the strings and the filaments where they show stretch and/or shrink. For the motion of the extensible string interacting with the external field the string changes its shape by stretch not to valance the elastic force with the external force. We also considered the motion of the stretched thin vortex filament and found exact solutions of the stretched vortex filament. The thin vortex filament is one of the fundamental modes in the fluid and is described by the localized induction equation (LIE). In this short note we would like to discuss two types of LIE for the vortex filament with and without stretch from a standpoint of similarity and difference. For this purpose we review how to derive LIE. The equations of motion for the incompressible and rotational inviscid fluid are given by