Geometry of the Kaup-Newell equation

It is known that the KdV equation appears naturally in the geometry of the orientation preserving diffeomorphic group Diff(S1). It is a geodesic flow of a L2 metric on the Bott-Virasoro group. The nonlinear Schrِdinger equation (NLSE) is an evolution equation analogous to the KdV equation which describes an isospectral deformation of the first order 2 × 2 matrix differential operator, yet the family of NSLE has not been studied via diffeomorphic groups. In this paper we derive the Kaup-Newell equation and its generalizations are the Euler-Poincaré flows on the space of first-order scalar (or matrix) differential operators. We show that the operators involved in the flow generated by the action of Vect(S1) are neither Poisson nor skew symmetric. We also discuss the relation between the KdV flow and the Kaup-Newell flow.