arXiv subjects

Thomas A. Ivey

Publications and source records attributed to Thomas A. Ivey.

22 records · Page 2Linked to original sources

arXiv2006

Finite-gap Solutions of the Vortex Filament Equation: Isoperiodic Deformations

We study the topology of quasiperiodic solutions of the vortex filament equation in a neighborhood of multiply covered circles. We construct these solutions by means of a sequence of isoperiodic deformations, at each step of which a real double point is "unpinched" to produce a new pair of branch points and therefore a solution of higher genus. We prove that every step in this process corresponds to a cabling operation on the previous curve, and we provide a labelling scheme that matches the deformation data with the knot type of the resulting filament.

nlin.SI↗

arXiv2002

Parametric Backlund transformations I: Phenomenology

We begin an exploration of parametric Backlund transformations for hyperbolic Monge-Ampere systems. We compute invariants for such transformations and explore the behavior of four examples regarding their invariants, symmetries, and conservation laws. We prove some preliminary results and indicate directions for further research.

math.AP↗

arXiv2001

An Inverse Problem from Sub-Riemannian Geometry

The geodesics for a sub-Riemannian metric on a three-dimensional contact manifold $M$ form a 1-parameter family of curves along each contact direction. However, a collection of such contact curves on $M$, locally equivalent to the solutions of a fourth-order ODE, are the geodesics of a sub-Riemannian metric only if a sequence of invariants vanish. The first of these, which was earlier identified by Fels, determines if the differential equation is variational. The next two determine if there is a well-defined metric on $M$ and if the given paths are its geodesics.

math.DG↗

arXiv1999

Knot types, homotopies and stability of closed elastic rods

The energy minimization problem associated to uniform, isotropic, linearly elastic rods leads to a geometric variational problem for the rod centerline, whose solutions include closed, knotted curves. We give a complete description of the space of closed and quasiperiodic solutions. The quasiperiodic curves are parametrized by a two-dimensional disc. The closed curves arise as a countable collection of one-parameter families, connecting the m-fold covered circle to the n-fold covered circle for any m,n relatively prime. Each family contains exactly one self-intersecting curve, one elastic curve, and one closed curve of constant torsion. Two torus knot types are represented in each family, and all torus knots are represented by elastic rod centerlines.

math.DG↗