Gauging of Geometric Actions and Integrable Hierarchies of KP Type

This work consist 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” representation of the pertinent equations of motion on the coadjoint orbit. Second, in the special case of G being 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 Ĝ. For Ĝ = ŜL(M +R) the latter hiearchies are equivalent to a class of constrained (reduced) KP hierarchies called cKPR,M , which contain as special cases a series of well-known integrable systems (mKdV, AKNS, Fordy-Kulish, Yajima-Oikawa etc.). We describe in some detail the loop algebras of additional (non-isospectral) symmetries of cKPR,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 cKPR,M hierarchies. Along the way we exhibit explicitly the interrelation between the Sato pseudo-differential operator formulation and the algebraic (generalized) Drinfeld-Sokolov formulation of cKPR,M hierarchies. Also we present the explicit derivation of the general Darboux-Bäcklund solutions of cKPR,M preserving their additional (non-isospectral) symmetries, which for R = 1 contain among themselves solutions to the gauged SL(M + 1)/U(1)× SL(M) WZNW field equations.