Related papers: The Multicomponent KP Hierarchy: Differential Fay …

We consider a general framework for integrable hierarchies in Lax form and derive certain universal equations from which `functional representations' of particular hierarchies (like KP, discrete KP, mKP, AKNS), i.e. formulations in terms of…

Using the bilinear formalism, we consider multicomponent and matrix modified KP hierarchies. The main tool is the bilinear identity for the tau-function which is realized as an expectation value of a Clifford group element composed from…

The discrete KP hierarchy is also known as the $(l-l')$--th modified KP hierarchy. Here in this paper, we consider the corresponding two--component generalization, called the two--component discrete KP (2dKP) hierarchy. Firstly, starting…

We give a detailed account of the N -component Toda lattice hierarchy. This hierarchy is an extended version of the one introduced by Ueno and Takasaki. Our version contains N discrete variables rather than one. We start from the Lax…

Starting from free charged fermions we give equivalent definitions of the $n\/$-component KP hierarchy, in terms of $\tau\/$-functions $\tau_\alpha\/$ (where $\alpha \in M =\/$ root lattice of $sl_n\/$), in terms of $n \times n\/$ matrix…

This, to a large extent, expository paper, describes the theory of multicomponent hierarchies of evolution equations of XKP type, where X=A, B, C or D, and AKP=KP, and their reductions, associated to the conjugacy classes of the Weyl groups…

A $q$-analogue of the tau function of the modified KP hierarchy is defined by a change of independent variables. This tau function satisfies a system of bilinear $q$-difference equations. These bilinear equations are translated to the…

Recently the study of Fay-type identities revealed some new features of the DKP hierarchy (also known as "the coupled KP hierarchy" and "the Pfaff lattice"). Those results are now extended to a Toda version of the DKP hierarchy (tentatively…

We review the notion of differential Fay identities and demonstrate, through case studies, its new role in integrable hierarchies of the KP type. These identities are known to be a convenient tool for deriving dispersionless Hirota…

With the square eigenfunctions symmetry constraint, we introduce a new extended matrix KP hierarchy and its Lax representation from the matrix KP hierarchy by adding a new $\tau_B$ flow. The extended KP hierarchy contains two time series…

In this paper, we prove the existence of tau functions of the discrete modified KP hierarchy and define the squared eigenfunction symmetry. Meanwhile, the Fay identity with its difference form, the squared eigenfunction potentials and the…

It is quite basic in integrable systems to deriving Lax equations from bilinear equations. For multi--component KP theory, corresponding Lax structures are mainly constructed by matrix pseudo-differential operators for fixed discrete…

Questions on random matrices and on non-intersecting Brownian motions have led to the study of moment matrices with regard to several weights. The purpose of this paper is to show that the determinants of such moment matrices satisfy, upon…

We find all formal solutions to the $\hbar$-dependent KP hierarchy. They are characterized by certain Cauchy-like data. The solutions are found in the form of formal series for the tau-function of the hierarchy and for its logarithm (the…

Some new formulas for the KP hierarchy are derived from the differential Fay identity. They proved to be useful for the $k$-constrained hierarchies providing a series of determinant identities for them. A differential equation is introduced…

The goal of this paper is to identify the universal Whitham hierarchy of genus zero with a dispersionless limit of the multi-component KP hierarchy. To this end, the multi-component KP hierarchy is (re)formulated to depend on several…

Restricting a linear system for the KP hierarchy to those independent variables t\_n with odd n, its compatibility (Zakharov-Shabat conditions) leads to the "odd KP hierarchy". The latter consists of pairs of equations for two dependent…

We revisit dispersionless version of the multicomponent KP hierarchy considered previously by Takasaki and Takebe. In contrast to their study, we do not fix any distinguished component treating all of them on equal footing. We obtain…

We construct a new multi-component CKP hierarchy based on the eigenfunction symmetry reduction. It contains two types of CKP equation with self-consistent sources which Lax representations are presented. Also it admits reductions to…

We use a Grassmannian framework to define multi-component tau functions as expectation values of certain multi-component Fermi operators satisfying simple bilinear commutation relations on Clifford algebra. The tau functions contain both…