Recent Developments in Integrable Curve Dynamics

The dynamics of vortex laments has provided for almost a century one of the most interesting connections between diierential geometry and integrable equations, and an example in which knotted curves arise as solutions of diierential equations possessing remarkably rich geometrical structures. In this paper we present several aspects of the integrable evolution of a closed vor-tex lament in an Eulerian uid. Starting with the derivation of the equations from an idealised physical model, we describe their hamiltonian formulation on an appropriate innnite-dimensional phase space. Then we discuss a transformation discovered by Hasimoto which converts the lament equation into the cubic nonlinear Schrr odinger equation, unveiling its complete integrability. We work principally with the evolution equation for the tangent indicatrix of the vortex lament, more suitably describing the dynamics of closed curves: this is also a completely integrable soliton equation, originally derived to model the evolution of a continuous Heisenberg spin chain. This paper has a double task. On the one hand it aims at providing an introduction to soliton equations in a concrete geometrical setting. For this purpose, we discuss an important technique for constructing a large class of special solutions, among which are some interesting torus knots. Also, we make use of the BB acklund transformation for the Heisenberg spin chain model to explore the symmetries of a given curve and to construct its homoclinic solutions. On the other hand, some results and techniques of soliton theory are revis-ited in the light of the diierential geometry of curves, where their geometric signiicance becomes transparent. In this context, we introduce a new or-thonormal framing of a given curve and formulate a geometric construction of the Hasimoto transformation. We use this approach to show that the relation between the cubic nonlinear Schrr odinger equation and the Heisenberg model is realised by a Poisson map between two diierent Poisson structures. 1.1. The physical model. Whirlpools and smoke rings are common examples of vortex laments: approximately one-dimensional regions where the velocity distribution of a uid has a rotational component. We give below an idealised description of the self-induced dynamics of a closed line vortex in a Eulerian uid, based on Batchelor's approach 4]. Let u be the velocity distribution of an incompressible (divu = 0) uid lling an unbounded region in space. We suppose that the vorticity w = curlu (measuring the rotational component of the velocity eld) is zero at points not …