Loop-Algebra and Virasoro Symmetries of Integrable Hierarchies of KP Type

Abstract

We propose a systematic treatment of symmetries of KP integrable systems, including constrained (reduced) KP models ${\sl cKP}_{R,M}$, and their multi-component (matrix) generalizations. Any such integrable hierarchy is shown to possess an additional $({\hat U}(1)\oplus{\hat {SL}}(M))_{+} \oplus ({\hat {SL}}(M+R))_{-}$ loop-algebra symmetry. Also we provide a systematic construction of the full algebra of Virasoro additional symmetries in the case of constrained KP models which requires a nontrivial modification of the known Orlov-Schulman construction for the general unconstrained KP hierarchy. Multi-component KP hierarchies are identified as ordinary (scalar) one-component KP hierarchies supplemented with the Cartan subalgebra of the additional symmetry algebra, which provides the basis of a new method for construction of soliton-like solutions. Davey-Stewartson and $N$-wave resonant systems arise as symmetry flows of ordinary ${\sl cKP}_{R,M}$ hierarchies.