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 e, the ratio of core radius to curvature radius, the velocity of a vortex filament is derived to O(ε{sup}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.
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