Diffeomorphisms on S¹, projective structures and integrable systems

In this paper we consider a projective connection as defined by the nth-order Adler-Gelfand-Dikii (AGO) operator on the circle. It is well-known that the Korteweg-de Vries (KdV) equation is the archetypal example of a scalar Lax equation defined by a Lax pair of scalar nth-order differential (AGD) operators. In this paper we derive (formally) the KdV equation as an evolution equation of the AGD operator (at least for n < 4) under the action of Vect(S¹). The solutions of the AGD operator define an immersion R → ℝP^n-1 in homogeneous coordinates. In this paper we derive the Schwarzian KdV equation as an evolution of the solution curve associated with Δ⁽ⁿ⁾, for n ≤ 4.