A fourth-order dispersive flow into K\"ahler manifolds

TL;DR

This paper proves a short-time existence theorem for a fourth-order dispersive flow of curves into K"ahler manifolds, generalizing physical models like vortex filament motion, using energy methods and gauge transforms.

Contribution

It introduces a new approach with a bounded gauge transform and local smoothing effects to establish existence for complex geometric dispersive flows.

Findings

Established short-time existence of solutions

Generalized physical models to K"ahler manifolds

Developed a novel energy method approach

Abstract

We discuss a short-time existence theorem of solutions to the initial value problem for a fourth-order dispersive flow for curves parametrized by the real line into a compact K\"ahler manifold. Our equations geometrically generalize a physical model describing the motion of a vortex filament or the continuum limit of the Heisenberg spin chain system. Our results are proved by using so-called the energy method. We introduce a bounded gauge transform on the pullback bundle, and make use of local smoothing effect of the dispersive flow a little.