Structure of a fourth-order dispersive flow equation through the generalized Hasimoto transformation

TL;DR

This paper introduces a generalized Hasimoto transformation to analyze a fourth-order nonlinear dispersive PDE for curve flows on Kähler manifolds, extending the Schrödinger flow to higher orders and dimensions.

Contribution

It develops a framework transforming the fourth-order dispersive PDE into a system of equations via a new generalized Hasimoto transformation applicable to higher-dimensional Kähler manifolds.

Findings

Derived explicit systems for specific Kähler manifolds

Demonstrated the transformation on complex Grassmannian

Extended the Schrödinger flow to fourth-order equations

Abstract

This paper focuses on a one-dimensional fourth-order nonlinear dispersive partial differential equation for curve flows on a K\"ahler manifold. The equation arises as a fourth-order extension of the one-dimensional Schr\"odinger flow equation, with physical and geometrical backgrounds. First, this paper presents a framework that can transform the equation into a system of fourth-order nonlinear dispersive partial differential-integral equations for complex-valued functions. This is achieved by developing the so-called generalized Hasimoto transformation, which enables us to handle general higher-dimensional compact K\"ahler manifolds. Second, this paper demonstrates the computations to obtain the explicit expression of the derived system for three examples of the compact K\"ahler manifolds, dealing with the complex Grassmannian as an example in detail.