Three-dimensional Curve Motions Induced by the Modified Korteweg-de Vries Equation

TL;DR

This paper constructs quasi-periodic solutions for curve motions induced by the mKdV equation, describing complex filament dynamics in fluid flows using elliptic functions, and classifies different filament shapes.

Contribution

It introduces explicit elliptic function solutions for the curve equation related to the mKdV, including various filament configurations and spectral classifications.

Findings

Solutions describe vortex filament dynamics in fluids.

Includes diverse filament shapes like Kelvin waves and solitons.

Classifies solutions into two spectral types.

Abstract

We have constructed one-phase quasi-periodic solutions of the curve equation induced by the mKdV equation. The solution is expressed in terms of the elliptic functions of Weierstrass. This solution can describe curve dynamics such as a vortex filament with axial velocity embedded in an incompressible inviscid fluid. There exist two types of curves (type-A, type-B) according to the form of the main spectra of the finite-band integrated solution. Our solution includes various filament shapes such as the Kelvin-type wave, the rigid vortex, plane curves, closed curves, and the Hasimoto one-solitonic filament.