Initial and initial-boundary value problems for a vortex f ilament with or without axial flow

The system of equations $\mathrm{x}_{\mathrm{t}}=_{\mathrm{X}\mathrm{s}}\cross \mathrm{x}_{\mathrm{S}}\mathrm{s}+\mathrm{a}\{\mathrm{x}_{\mathrm{S}\epsilon \mathrm{S}}+(3/2)\mathrm{x}_{\mathrm{s}\epsilon}\cross(\mathrm{X}_{\mathrm{s}\mathrm{s}\mathrm{s}}\cross \mathrm{X})\} $ (1. 1) approximately describes the deformation of a vortex $\mathrm{f} $ ilament with or without axial velocity in its thin core, in a perfect fluid. Her