Geometric Schr\"odinger-Airy Flows on K\"ahler Manifolds

TL;DR

This paper introduces a new class of geometric flows on K"ahler manifolds that unify various physical and mathematical models, and establishes local and global existence results for these flows.

Contribution

It defines the Geometric Schr"odinger-Airy flows on K"ahler manifolds and proves their existence, connecting multiple models in physics and mathematics.

Findings

Defined a new class of geometric flows on K"ahler manifolds.

Proved local and global existence results for these flows.

Unified several physical and mathematical models under this framework.

Abstract

We define a class of geometric flows on a complete K\"ahler manifold to unify some physical and mechanical models such as the motion equations of vortex filament, complex-valued mKdV equations, derivative nonlinear Schr\"odinger equations etc. Furthermore, we consider the existence for these flows from $S^1$ into a complete K\"ahler manifold and prove some local and global existence results.