Motion of a curved vortex filament: Higher-order asymptotics

Three-dimensional motion of a slender vortex tube, embedded in an inviscid incompressible fluid, is investigated based on the Euler equations. Using the method of matched asymptotic expansions in a small parameter is an element of, the ratio of core radius to curvature radius, the velocity of a vortex filament is derived to O(is an element of (3)), whereby the influence of elliptical deformation of the core due to the self-induced strain is taken into account. In the localized induction approximation, this is reducible to a completely integrable evolution equation among the localized induction hierarchy.