Structure of a Fourth-Order Dispersive Flow Equation Through the Generalized Hasimoto Transformation

This paper focuses on a one-dimensional fourth-order nonlinear dispersive partial differential equation for curve flows on a K & auml;hler manifold. The equation arises as a fourth-order extension of the one-dimensional Schr & ouml;dinger 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 & auml;hler manifolds. Second, this paper demonstrates the computations to obtain the explicit expression of the derived system for three examples of the compact K & auml;hler manifolds, dealing with the complex Grassmannian as an example in detail. In particular, the result of the computations when the manifold is a Riemann surface or a complex Grassmannian verifies that the expression of the system derived by our framework actually unifies the ones derived previously. Additionally, the computation when the compact K & auml;hler manifold has a constant holomorphic sectional curvature, the setting of which has not been investigated, is also demonstrated.