Covariant differential operators for densities and KdV equation as a flow on diffeomorphism groups (Q1826096)

In the main statements of this paper the author has presented an algorithm for constructing differential operators of the form \[ D_{\hat q}^{(k)}=(d/d\theta)^ k+\sum^{k-2}_{j=0}A^ k_ j(q)(\theta)(d/d\theta \quad)^ j,\quad k=2,3,4,..., \] such that \(D_{\hat q}^{(k)}: \Omega_{(1-k)/2}(S^ 1)\to \Omega_{(1+k)/2}(S^ 1),\) where \(\Omega_ u(S^ 1)\) is the space of densities of weight u, and \(D_{\hat q}^{(k)}\) has properties similar to those of Hill's operator \(D_{\hat q}^{(2)}=(d/d\theta)^ 2+q(\theta).\) The introduction of the paper contains an indication of the connections of the equations with quantization in string theory, and further considerations indicate connections with the KdV equation \(\dot q_ t=q_ t^{(3)}+3q_ tq'_ t,\) which occurs as a necessary condition for the existence of a solution of \[ D_{\hat q}^{(2)}\psi_ t=0;\quad {\dot \psi}_ t=-v'_ t\psi_ t+v_ t\psi '_ t. \]