Constrained KP Models as Integrable Matrix Hierarchies

Abstract

We formulate the constrained KP hierarchy (denoted by \cKP${}_{K+1,M}$) as an affine $\widehat{sl}(M+K+1)$ matrix integrable hierarchy generalizing the Drinfeld-Sokolov hierarchy. Using an algebraic approach, including the graded structure of the generalized Drinfeld-Sokolov hierarchy, we are able to find several new universal results valid for the \cKP hierarchy. In particular, our method yields a closed expression for the second bracket obtained through Dirac reduction of any untwisted affine Kac-Moody current algebra. An explicit example is given for the case $\widehat{sl}(M+K+1)$, for which a closed expression for the general recursion operator is also obtained. We show how isospectral flows are characterized and grouped according to the semisimple {\em non-regular} element $E$ of $sl(M+K+1)$ and the content of the center of the kernel of $E$.

LaTeX, 19 pgs

References in corpus (2)

• Tau-functions and Dressing Transformations for Zero-Curvature Affine Integrable Equations
• Dispersionful analogues of Benney's equations and $N$-wave systems

Cited by in corpus (17)

• Tau-functions and Dressing Transformations for Zero-Curvature Affine Integrable Equations
• q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy
• Dispersionful analogues of Benney's equations and $N$-wave systems
• An analytic description of the vector constrained KP hierarchy
• Integrable Hierarchy for Multidimensional Toda Equations and Topological-Anti-topological Fusion
• The complex Sine-Gordon equation as a symmetry flow of the AKNS Hierarchy
• Tau-Functions generating the Conservation Laws for Generalized Integrable Hierarchies of KdV and Affine-Toda type
• The vector k-constrained KP hierarchy and Sato's Grassmannian
• Solitons from Dressing in an Algebraic Approach to the Constrained KP Hierarchy
• Generalized Backlund transformations for Affine Toda Hierarchies
• Vertex Operators and Solitons of Constrained KP Hierarchies
• Gauging of Geometric Actions and Integrable Hierarchies of KP Type
• On the Miura and Backlund transformations associated with the supersymmetric Gelfand-Dickey bracket
• Matrix Formulation of Hamiltonian Structures of Constrained KP Hierarchy
• Extended matrix Gelfand-Dickey hierarchies: reduction to classical Lie algebras
• Soliton equations related to the affine Kac-Moody algebra D^(1)_4
• The Hamiltonian Structure of Soliton Equations and Deformed W-Algebras