GAUGING OF GEOMETRIC ACTIONS AND INTEGRABLE HIERARCHIES OF THE KP TYPE

Abstract

This work consists of two interrelated parts. First, we derive massive gauge-invariant generalizations of geometric actions on coadjoint orbits of arbitrary (infinite-dimensional) groups G with central extensions, with gauge group H being certain (infinite-dimensional) subgroup of G. We show that there exist generalized "zero-curvature" representations of the pertinent equations of motion on the coadjoint orbit. Second, in the special case of G being the Kac–Moody group, the equations of motion of the underlying gauged WZNW geometric action are identified as additional-symmetry flows of generalized Drinfeld–Sokolov integrable hierarchies based on the loop algebra [Formula: see text]. For [Formula: see text] the latter hierarchies are equivalent to the class c KP R,M of constrained (reduced) KP hierarchies. We describe in some detail the loop algebras of additional (nonisospectral) symmetries of c KP R,M hierarchies. Apart from gauged WZNW models, certain higher-dimensional nonlinear systems such as Davey–Stewartson and N-wave resonant systems are also identified as additional symmetry flows of c KP R,M hierarchies. Also we present the explicit derivation of the general Darboux–Bäcklund solutions of c KP R,M hierarchies preserving their additional (nonisospectral) symmetries, which for R=1 contain among themselves solutions to the gauged SL (M+1)/ U (1)× SL (M) WZNW field equations.