Local existence of a fourth order dispersive curve flow on locally hermitian symmetric spaces and the application

TL;DR

This paper proves local existence of solutions for a fourth order dispersive curve flow on compact locally hermitian symmetric spaces, using geometric energy methods and gauge transformations, with applications to generalized bi-Schr"odinger flows.

Contribution

It establishes local well-posedness for a complex geometric PDE on specific symmetric spaces, extending the understanding of dispersive curve flows.

Findings

Local existence of solutions on compact locally hermitian symmetric spaces.

Application to generalized bi-Schr"odinger flow equations.

Use of geometric energy method with gauge transformation.

Abstract

This paper is concerned with a fourth order nonlinear dispersive partial differential equation for closed curve flow on a K\"ahler manifold. The main results is that the initial value problem has a solution locally in time if the K\"ahler manifold is a compact locally hermitian symmetric space. The proof is based on the geometric energy method combined with a nice gauge transformation to eliminate the loss of derivatives. Interestingly, the results can be applied to construct a generalized bi-Schr\"odinger flow proposed by Ding and Wang. The assumption on the manifold plays a crucial role both to enjoy a good solvable structure of the problem and to reduce the generalized bi-Schr\"odinger flow equation to the one considered in the present paper.