Solutions of string, vortex, and anyon types for the hierarchy of the nonlinear Schrödinger equation

We consider the relation that associates an infinite curve evolving in the $E_3$ space with each equation of the hierarchy of the nonlinear Schrödinger and the modified Korteveg–de Vries equations. We show that the low levels of the hierarchy correspond to known objects, which are strings and vortex lines endowed with various structures. We consider one of the hierarchy levels corresponding to the dynamics of the vortex line in the local induction approximation in detail. We construct the Hamiltonian description of the corresponding dynamics admitting an interpretation in terms of a quasiparticle in a plane, the "anyon." We propose a scheme for quantizing the theory in a framework in which we obtain a formula for a (generally fractional) spin value.