J. Harnad

Dual isomonodromic deformations and moment maps to loop algebras

The Hamiltonian structure of the monodromy preserving deformation equations of Jimboet al [JMMS] is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras. The nonautonomous Hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on Poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those determining the moment maps. This construction is shown to lead to “dual” pairs of matrix differential operators whose monodromy is preserved under the same family of deformations. As illustrative examples, involving discrete and continuous reductions, a higher rank generalization of the Hamiltonian equations governing the correlation functions for an impenetrable Bose gas is obtained, as well as dual pairs of isomonodromy representations for the equations of the Painlevé transcendentsPV andVI.

Duality, Biorthogonal Polynomials¶and Multi-Matrix Models

The statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized Christoffel–Darboux form constructed from sequences of biorthogonal polynomials. For measures involving exponentials of a pair of polynomials V1, V2 in two different variables, these kernels may be expressed in terms of finite dimensional “windows” spanned by finite subsequences having length equal to the degree of one or the other of the polynomials V1, V2. The vectors formed by such subsequences satisfy “dual pairs” of first order systems of linear differential equations with polynomial coefficients, having rank equal to one of the degrees of V1 or V2 and degree equal to the other. They also satisfy recursion relations connecting the consecutive windows, and deformation equations, determining how they change under variations in the coefficients of the polynomials V1 and V2. Viewed as overdetermined systems of linear difference-differential-deformation equations, these are shown to be compatible, and hence to admit simultaneous fundamental systems of solutions. The main result is the demonstration of a spectral duality property; namely, that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters.

Superposition principles for matrix Riccati equations

A superposition rule is obtained for the matrix Riccati equation (MRE) W=A+WB+CW+WDW [where W(t), A(t), B(t), C(t), and D(t) are real n×n matrices], expressing the general solution in terms of five known solutions for all n≥2. The symplectic MRE (W=WT, A=AT, D=DT, C=BT) is treated separately, and a superposition rule is derived involving only four known solutions. For the ‘‘unitary’’ and GL(n,R) subcases (with D=A and C=BT, or D=−A and C=BT, respectively), superposition rules are obtained involving only two solutions. The derivation of these results is based upon an interpretation of the MRE in terms of the action of the groups SL(2n,R), SP(2n,R), U(n), and GL(n,R) on the Grassman manifold Gn(R2n).

Backlund Transformations for Nonlinear

This work deals with Bäcklund transformations for the principal SL(n, ℂ) sigma model together with all reduced models with values in Riemannian symmetric spaces. First, the dressing method of Zakharov, Mikhailov, and Shabat is shown, for the case of a meromorphic dressing matrix, to be equivalent to a Bäcklund transformation for an associated, linearly extended system. Comparison of this multi-Bäcklund transformation with the composition of ordinary ones leads to a new proof of the permutability theorem. A new method of solution for such multi-Bäcklund transformations (MBT) is developed, by the introduction of a “soliton correlation matrix” which satisfies a Riccati system equivalent to the MBT. Using the geometric structure of this system, a linearization is achieved, leading to a nonlinear superposition formula expressing the solution explicitly in terms of solutions of a single Bäcklund transformation through purely linear algebraic relations. A systematic study of all reductions of the system by involutive automorphisms is made, thereby defining the multi-Bäcklund transformations and their solution for all Riemannian symmetric spaces.

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AbstractWe consider biorthogonal polynomials that arise in the study of a generalization of two–matrix Hermitian models with two polynomial potentials V1(x), V2(y) of any degree, with arbitrary complex coefficients. Finite consecutive subsequences of biorthogonal polynomials (‘‘windows’’), of lengths equal to the degrees of the potentials V1 and V2, satisfy systems of ODE’s with polynomial coefficients as well as PDE’s (deformation equations) with respect to the coefficients of the potentials and recursion relations connecting consecutive windows. A compatible sequence of fundamental systems of solutions is constructed for these equations. The (Stokes) sectorial asymptotics of these fundamental systems are derived through saddle-point integration and the Riemann-Hilbert problem characterizing the differential equations is deduced.

Models With Values in Riemannian Symmetric Spaces

Abstract: The Fredholm determinants of a special class of integrable integral operators K supported on the union of m curve segments in the complex λ-plane are shown to be the τ-functions of an isomonodromic family of meromorphic covariant derivative operators , having regular singular points at the 2m endpoints of the curve segments, and a singular point of Poincaré index 1 at infinity. The rank r of the corresponding vector bundle over the Riemann sphere equals the number of distinct terms in the exponential sum defining the numerator of the integral kernel. The matrix Riemann–Hilbert problem method is used to deduce an identification of the Fredholm determinant as a τ-function in the sense of Segal–Wilson and Sato, i.e., in terms of abelian group actions on the determinant line bundle over a loop space Grassmannian. An associated dual isomonodromic family of covariant derivative operators , having rank n= 2m, and r finite regular singular points located at the values of the exponents defining the kernel of K is derived. The deformation equations for this family are shown to follow from an associated dual set of Riemann–Hilbert data, in which the rôles of the r exponential factors in the kernel and the 2m endpoints of its support are interchanged. The operators are analogously associated to an integral operator whose Fredholm determinant is equal to that of K.

Differential Systems for Biorthogonal Polynomials Appearing in 2-Matrix Models and the Associated Riemann–Hilbert Problem

Abstract This work is concerned with the derivation of superposition rules which express the general solution of ordinary differential equations. x = η(x,t). (x, η ϵ R n , t ϵ R ) . in terms of a finite number of particular solutions. The point of departure is Lies criterion according to which such a rule exists if and only if the vector fields η ( x , t ). ∇ generate a finite dimensional Lie algebra. We provide three different constructive methods for deriving superposition rules and apply them to systems of coupled Riccati equations of the projective and conformal types based, respectively, on the Lie algebra sl ( n + 1, R ) and o ( p + 1, n − p + 1).

Integrable Fredholm Operators and Dual Isomonodromic Deformations

The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such measures. These are shown to preserve the generalized monodromy of the associated rank-2 rational covariant derivative operators. The corresponding matrix models, consisting of unitarily diagonalizable matrices with spectra supported on these contours are analyzed, and it is shown that all coefficients of the associated spectral curves are given by logarithmic derivatives of the partition function or, more generally, the gap probabilities. The associated isomonodromic tau functions are shown to coincide, within an explicitly computed factor, with these partition functions.

Systems of ordinary differential equations with nonlinear superposition principles

For one-matrix models with polynomial potentials, the explicit relationship between the partition function and the isomonodromic tau function for the 2 × 2 polynomial differential systems satisfied by the associated orthogonal polynomials is derived.

Semiclassical Orthogonal Polynomials, Matrix Models and Isomonodromic Tau Functions

Generalized Drinfeld-Sokolov (DS) hierarchies are constructed through local reductions of Hamiltonian flows generated by monodromy invariants on the dual of a loop algebra. Following earlier work of De Groot et al., reductions based upon graded regular elements of arbitrary Heisenberg subalgebras are considered. We show that, in the case of the nontwisted loop algebra ℓ(gln), graded regular elements exist only in those Heisenberg subalgebras which correspond either to the partitions ofn into the sum of equal numbersn=pr or to equal numbers plus onen=pr+1. We prove that the reduction belonging to the grade 1 regular elements in the casen=pr yields thep×p matrix version of the Gelfand-Dickeyr-KdV hierarchy, generalizing the scalar casep=1 considered by DS. The methods of DS are utilized throughout the analysis, but formulating the reduction entirely within the Hamiltonian framework provided by the classical r-matrix approach leads to some simplifications even forp=1.