A number of important model linear dispersive equations give rise to
interesting curve flows in differential geometry. In this talk we will
discuss some of these including the Schrodinger curve flow on the two
sphere, the Hodge star mean curvature curve flow in Euclidean and
Lorentzian 3-space, and the geometric Airy curve flow on Euclidean
space and the affine space. The equations of curvatures of these
curve flows turn out to be soliton equations. Hence we can use
techniques from soliton theory to study these curve flows. In
particular, we can construct infinitely many families of explicit
solutions and solve the periodic Cauchy problem.