Orthogonal and Symplectic Matrix Integrals and Coupled KP Hierarchy

Abstract

Orthogonal and symplectic matrix integrals are investigated. It is shown that the matrix integrals can be considered as a τ -function of the coupled KP hierarchy, whose solution can be expressed in terms of pfaffians.Over the past decade, the intimate relationships between matrix integrals and nonlinear integrable systems have been clarified, particularly in the context of string theory. In such cases, nonperturbative properties of physical quantities can be evaluated by the use of the integrable structures of the models. (For review, see refs. 1-4.)Here, we consider a matrix integral over an ensemble of Hermitian matriceswhere β = 1, 2 or 4, for which the ensemble is, respectively, orthogonal, unitary or symplectic and η(x, t) = ∞ m=1 x m t m . An interesting observation is that, when β = 2, the quantity Z (β=2) N {t} is a τ -function of the Kadomtsev-Petviashvili (KP) hierarchy [2,3,4,5]. In fact, the integral (1) can be identified as a continuum limit of the multisoliton solution of the hierarchy. This fact may be easily observed if we rewrite Z (β=2) N in the following form:It may be also clear that the quantity Z (β=2) N satisfies the bilinear Toda equationHere, we give other examples of such kinds of relationship. The orthogonal case β = 1 and the symplectic case β = 4 lead to τ -functions of the coupled KP hierarchy proposed

On a Generalized Tzitzeica Equation

“…Because of the resemblance of the first equation in (18) to the bilinear KP equation -it is of the form 'KP + source term' -the system (18) is often referred to as the coupled KP equation (in Hirota form). It can be written as a set of 2 + 1 dimensional evolution equations, through the dependent variable transformation Recently there has been renewed interest in the above system of equations and its associated hierarchy, mainly in connection with the study of matrix integrals [15] and orthogonal polynomials [1,2], but also because of the remarkable interaction properties of its soliton solutions [10]. It is known that the coupled KP hierarchy is connected to an infinite rank Kac-Moody algebra of D-type [14,16]; more precisely, it can be shown [13,17] that the coupled KP hierarchy is associated to an algebra (denoted as D ∞ ) isomorphic to the usual D ∞ algebra.…”

Matrix Integrals and Several Integrable Differential-Difference Systems

“…Over the past decade, many remarkable and excellent results have been achieved towards clarifying close relations between integrals of the form (1) with β = 1, 2, 4 and nonlinear integrable systems (see, e.g. [5]- [10]). An important step involved to induce these integrable equations is to insert time-parameters into the integrals.…”

“…We now turn to establish the relation between the matrix integral (2) with β = 1, 4 and the Pfaffianized differential-difference KP system (8)- (10). In order to do so, we need the following formulae due to de Bruijn [17]:…”

“…In order to do so, we need the formulae (22)-(23) due to de Bruijn [17] again. Similar to [10], we can rewrite Z …”

Pfaff $\tau$ -functions

“…The Pfaff lattice appears implicitly in the work of Jimbo and Miwa as one half of the D ∞ -hierarchy (compare ( 0.7) (or (3.2)) with the case l = l of formula (7.7) in [15]), in the work of Hirota et al, in the context of the coupled KP hierarchy (compare, e.g., (0.5) and (0.9) with formulas (3.5) and (3.25a) in [13], respectively), in the work of Kac and van de Leur [16] in the context of the DKP hierarchy (on the exact connection, see forthcoming work by J. van de Leur [24]), and in the recent work of S. Kakei [17,18], who realized Hirota et al's coupled KP hierarchy as a restriction of the 2-component KP hierarchy instead of the 2-Toda lattice, and studied its relation to matrix integrals among other aspects.…”