The multicomponent 2D Toda hierarchy is analyzed through a factorization problem associated to an infinitedimensional group. A new set of discrete flows is considered and the corresponding Lax and Zakharov-Shabat equations are characterized. Reductions of block Toeplitz and Hankel bi-infinite matrix types are proposed and studied. Orlov-Schulman operators, string equations and additional symmetries (discrete and continuous) are considered. The continuous-discrete Lax equations are shown to be equivalent to a factorization problem as well as to a set of string equations. A congruence method to derive site independent equations is presented and used to derive equations in the discrete multicomponent KP sector (and also for its modification) of the theory as well as dispersive Whitham equations.
IntroductionThis paper revisits the multicomponent 2D Toda hierarchy [30] from the point of view of the factorization problem associated to an infinite-dimensional group. Our main motivation is the recent discovery [3] of underlying integrable structures of multicomponent type in the theory of multiple orthogonal polynomials which is in turn connected to models of non-intersecting Brownian motions. Having in mind the fruitful applications of the Toda hierarchy to the theory of orthogonal polynomials and to the Hermitian random matrix model (see for instance [14]-[21]), it is expected that the formalism of multicomponent integrable hierarchies can be similarly applied to the study and characterization of multiple orthogonal polynomials and non-intersecting Brownian motions. In particular, the semiclassical (dispersionless) limit of multicomponent integrable hierarchies should be relevant for the analysis of large N ) type limits, see for instance [22]. An important piece of the technique required for these applications was recently provided by Takasaki and Takebe [27,28]. Indeed, they proved that the universal Whitham hierarchy (genus 0 case) [16] can be obtained as a particular dispersionless limit of the multicomponent KP hierarchy.The applications of the Toda hierarchy to the characterization of semiclassical limits make an essential use of the notion of string equations [14]-[21]- [10]. In recent years the formalism of string equations for dispersionless integrable hierarchies [26] has been much developed [32,19] but, to our knowledge, a similar formalism for dispersive multicomponent integrable hierarchies is not yet available. One of the main goals of this paper is to extend the formalism of string equations to multicomponent 2D Toda hierarchies. In this sense the consideration of factorization problems for these hierarchies turns to be of great help in order to introduce basic ingredients such as discrete flows, Orlov-Schulman operators and additional symmetries.The theory of the multicomponent KP hierarchy is discussed in length in the papers [15,4], see also [20] for its applications to geometric nets of conjugate type. In [30] it was noticed that τ functions of a 2N -multicomponent KP provide solutions of the N -mul...
