The KP equation has a large family of quasiperiodic multiphase solutions. These solutions can be expressed in terms of Riemann-theta functions. In this paper, a finite-dimensional 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 initial-value 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 Korteweg-de 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...

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 τ-functions of the KP hierarchy as a sum. In the same way, the new functions introduced here (n> 1) are used to expand quotients of τ-functions 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 τ-functions 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 τ-functions have recently played a prominent role in several papers on bispectrality [3, 7], Darboux transformations [2, 8] and random matrices [1]. In