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nSCHUR FUNCTIONS AND DETERMINANTS ON AN INFINITE GRASSMANNIAN
, 1998
"... Abstract. A set of functions is defined which is indexed by a positive integer n and partitions of integers. The case n = 1 reproduces the standard Schur polynomials. These functions are seen to arise naturally as a determinant of an action on the frame bundle of an infinite grassmannian. This fact ..."
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Abstract. A set of functions is defined which is indexed by a positive integer n and partitions of integers. The case n = 1 reproduces the standard Schur polynomials. These functions are seen to arise naturally as a determinant of an action on the frame bundle of an infinite grassmannian. This fact is well known in the case of the Schur polynomials (n = 1) and has been used to decompose the τfunctions of the KP hierarchy as a sum. In the same way, the new functions introduced here (n> 1) are used to expand quotients of τfunctions as a sum with Plücker coordinates as coefficients. Among their many important properties, the Schur polynomials [10] arise naturally as the determinant of an exponential function acting on the frame bundle of the grassmannian of the Hilbert space H = L 2 (S 1, C) [11]. It is for this reason that the τfunctions of the KP hierarchy can be expanded as a sum of Schur polynomials with the Plücker coordinates as coefficients [14, 15] (cf. [4]). Quotients of τfunctions have recently played a prominent role in several papers on bispectrality [3, 7], Darboux transformations [2, 8] and random matrices [1]. In
Canonical Variables for multiphase solutions of the KP equation
, 1998
"... The KP equation has a large family of quasiperiodic multiphase solutions. These solutions can be expressed in terms of Riemanntheta functions. In this paper, a finitedimensional canonical Hamiltonian system depending on a finite number of parameters is given for the description of each such soluti ..."
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The KP equation has a large family of quasiperiodic multiphase solutions. These solutions can be expressed in terms of Riemanntheta functions. In this paper, a finitedimensional canonical Hamiltonian system depending on a finite number of parameters is given for the description of each such solution. The Hamiltonian systems are completely integrable in the sense of Liouville. In effect, this provides a solution of the initialvalue problem for the thetafunction solutions. Some consequences of this approach are discussed. 1 Introduction In 1970, Kadomtsev and Petviashvili [1] derived two equations as generalizations of the Kortewegde Vries (KdV) equation to two spatial dimensions: (\Gamma4u t + 6uu x + u xxx ) x + 3oe 2 u yy = 0; (KP) where oe 2 = \Sigma1 and the subscripts denote differentiation. Depending on the physical situation, one derives the equation either with oe 2 = \Gamma1 or oe 2 = +1. The resulting partial differential equations are referred to as (KP1) and (KP...
INTEGRABLE SYSTEMS AND RANK ONE CONDITIONS FOR RECTANGULAR MATRICES
, 2004
"... Abstract. We provide a determinantal formula for taufunctions of the KP hierarchy in terms of rectangular, constant matrices A, B and C satisfying a rank one condition. This result is shown to generalize and unify many previous results of different authors on constructions of taufunctions for diff ..."
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Abstract. We provide a determinantal formula for taufunctions of the KP hierarchy in terms of rectangular, constant matrices A, B and C satisfying a rank one condition. This result is shown to generalize and unify many previous results of different authors on constructions of taufunctions for differential and difference integrable systems from square matrices satisfying rank one conditions. In particular, it contains as explicit special cases the formula of Wilson for taufunctions of rational KP solutions in terms of CalogeroMoser Lax matrices as well as our previous formula for KP tau functions in terms of almostintertwining matrices. 1.
Commuting Differential Operators of Rank 3 . . .
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2012
"... In this paper, we construct some examples of commuting differential operators L1 and L2 with rational coefficients of rank 3 corresponding to a curve of genus 2. ..."
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In this paper, we construct some examples of commuting differential operators L1 and L2 with rational coefficients of rank 3 corresponding to a curve of genus 2.