Invariant Differential Equations and the Adler-Gel'fand-Dikii Bracket

Abstract

In this paper we find an explicit formula for the most general vector evolution of curves on $RP^{n-1}$ invariant under the projective action of $SL(n,R)$. When this formula is applied to the projectivization of solution curves of scalar Lax operators with periodic coefficients, one obtains a corresponding evolution in the space of such operators. We conjecture that this evolution is identical to the second KdV Hamiltonian evolution under appropriate conditions. These conditions give a Hamiltonian interpretation of general vector differential invariants for the projective action of $SL(n,R)$, namely, the $SL(n,R)$ invariant evolution can be written so that a general vector differential invariant corresponds to the Hamiltonian pseudo-differential operator. We find common coordinates and simplify both evolutions so that one can attempt to prove the equivalence for arbitrary $n$.