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 $\widehat{\text{cG}}$. For $\widehat{\text{cG}}=\widehat{SL}(M+R)$ the latter hiearchies are equivalent to a class of constrained (reduced) KP hierarchies called ${\text{sl}cKP}_{R,M}$, 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 ${\text{sl}cKP}_{R,M}$ 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 ${\text{sl}cKP}_{R,M}$ hierarchies. Along the way we exhibit explicitly the interrelation between the Sato pseudo-differential operator formulation and the algebraic (generalized) Drinfeld-Sokolov formulation of ${\text{sl}cKP}_{R,M}$ hierarchies. Also we present the explicit derivation of the general Darboux-B"acklund solutions of ${\text{sl}cKP}_{R,M}$ preserving their additional (non-isospectral) symmetries, which for R=1 contain among themselves solutions to the gauged $SL(M+1)\mathrm{/}U(1)\times SL(M)$ WZNW field equations.