  The Hodge star mean curvature flow on a 3-dimension Riemannian or
pseudo-Riemannian manifold, the geometric Airy flow on a Riemannian manifold,
the Schrodingier flow on Hermitian manifolds, and the shape operator curve flow
on submanifolds are natural non-linear dispersive curve flows in geometric
analysis. A curve flow is integrable if the evolution equation of the local
differential invariants of a solution of the curve flow is a soliton equation.
For example, the Hodge star mean curvature flow on $R^3$ and on $R^{2,1}$, the
geometric Airy flow on $R^n$, the Schrodingier flow on compact Hermitian
symmetric spaces, and the shape operator curve flow on an Adjoint orbit of a
compact Lie group are integrable. In this paper, we give a survey of these
results, describe a systematic method to construct integrable curve flows from
Lax pairs of soliton equations, and discuss the Hamiltonian aspect and the
Cauchy problem of these curve flows.
