We construct a sequence of commuting central affine curve flows on $R^nbackslash 0$ invariant under the action of $SL(n,R)$ and prove the following results: (a) The central affine curvatures of a solution of the j-th central affine curve flow is a solution of the j-th flow of Gelfand-Dickey (GD$_n$) hierarchy on the space of n-th order differential operators. (b) We use the solution of the Cauchy problems of the GD$_n$ flow to solve the Cauchy problems for the central affine curve flows with periodic initial data and also with initial data whose central affine curvatures are rapidly decaying. (c) We obtain a bi-Hamiltonian structure for the central affine curve flow hierarchy and prove that it arises naturally from the Poisson structures of certain co-adjoint orbits. (d) We construct Backlund transformations, infinitely many families of explicit solutions and give a permutability formula for these curve flows.
We give the following results for Pinkalls central affine curve flow on the plane: (i) a systematic and simple way to construct the known higher commuting curve flows, conservation laws, and a bi-Hamiltonian structure, (ii) Baecklund transformations and a permutability formula, (iii) infinitely many families of explicit solutions. We also solve the Cauchy problem for periodic initial data.
