Integrable Equations and Motions of Plane Curves

The connection between motion of space or plane curves and integrable equations has drawn wide interest in the past and many results have been obtained. The pioneering work is due to Hasimoto where he showed in [1] that the nonlinear Schrödinger equation describes the motion of an isolated non-stretching thin vortex filament. Lamb [2] used the Hasimoto transformation to connect other motions of curves to the mKdV and sine-Gordon equations. Lakshmanan [3] related the Heisenberg spin model to the motion of space curves in the Euclidean space. Langer and Perline [4] obtained the Schrödinger heirarchy from motions of the non-stretching thin vortex filament. Motions of curves in S2 and S3 were considered by Doliwa and Santini [5]. Nakayama [6, 7] investigated motions of curves in Minkowski space and obtained the Regge– Lund equation, a couple of systems of the KdV equations and their hyperbolic type. In contrast to the motions of curves in space, only two types of integrable equations have been shown to be associated to motions of plane curves. In fact, Goldstein and Petrich [8] discovered that the dynamics of a non-stretching string on the plane produces the recursion operator of the mKdV hierarchy. Nakayama, Segur and Wadati [9] obtained the sine-Gordon equation by considering a nonlocal motion. They also pointed out that the Serret–Frenet equations for curves in E2 and E3 are equivalent to the AKNS-ZS spectral problem without spectral parameter [10, 11]. It is commonly believed that the KdV equation does not occur in the motion of plane curves. The purpose of this paper is to study motions of plane curves in Klein geometries. These geometries are characterized by their associated Lie algebras of vector fields in E2. We shall see that the KdV, Harry–Dym and Sawada–Kotera hierarchies and the Kaup–Kupershmidt equation naturally arise from the motions of plane curves in affine, centro-affine and similarity geometries. The outline of this paper is as follows. In Section 2, we give a brief discussion on the Klein geometry. In Sections 3, 4, and 5, we discuss motion laws of plane curves respectively in affine, centro-affine and similarity geometries. Section 7 is concluding remarks about this work.