Abstract
It is known that the KdV equation appears naturally in the geometry of the orientation preserving diffeomorphic group Diff( S 1). It is a geodesic flow of a L 2 metric on the Bott-Virasoro group. The nonlinear Schrödinger 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( S 1) are neither Poisson nor skew symmetric. We also discuss the relation between the KdV flow and the Kaup-Newell flow.
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