Abstract
It is known that the Korteweg–de Vries (KdV) equation is a geodesic flow of an L 2 metric on the Bott–Virasoro group. This can also be interpreted as a flow on the space of projective connections on S 1. The space of differential operators Δ ( n) = ∂ n + u 2 ∂ n−2 +⋯+ u n form the space of extended or generalized projective connections. If a projective connection is factorizable Δ ( n) =( ∂−(( n+1)/2−1) p 1)⋯( ∂+( n−1)/2 p n ) with respect to quasi primary fields p i ’s, then these fields satisfy ∑ i=1 n (( n+1)/2− i) p i =0. In this paper we discuss the factorization of projective connection in terms of affine connections. It is shown that the Burgers equation and derivative non-linear Schrödinger (DNLS) equation or the Kaup–Newell equation is the Euler–Arnold flow on the space of affine connections.
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