Generalized Hasimoto Transform of One-Dimensional Dispersive Flows into Compact Riemann Surfaces

TL;DR

This paper introduces a generalized Hasimoto transform for one-dimensional dispersive flows into compact Riemann surfaces, simplifying complex geometric PDEs to more manageable forms and linking them to linear dispersive PDE theory.

Contribution

It develops a new geometric transform that reduces dispersive flow equations into complex-valued equations, extending previous models to more general Riemann surfaces.

Findings

Constructed a good moving frame for the transform

Reduced geometric PDEs to complex-valued equations

Linked geometric reductions to linear dispersive PDE theory

Abstract

We study the structure of differential equations of one-dimensional dispersive flows into compact Riemann surfaces. These equations geometrically generalize two-sphere valued systems modeling the motion of vortex filament. We define a generalized Hasimoto transform by constructing a good moving frame, and reduce the equation with values in the induced bundle to a complex valued equation which is easy to handle. We also discuss the relationship between our reduction and the theory of linear dispersive partial differential equations.