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

TL;DR

This paper derives a general invariant vector evolution formula for curves on projective space under SL(n,R) action, linking it to KdV evolution and Hamiltonian structures, with potential for broad mathematical applications.

Contribution

It provides an explicit formula for invariant vector evolutions on projective space and connects these to Hamiltonian structures and integrable systems like KdV.

Findings

Derived explicit invariant vector evolution formula.

Linked the evolution to the second KdV Hamiltonian evolution.

Identified Hamiltonian interpretation of vector differential invariants.

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 $n$.