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

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.