An Algebraic Characterization of the Bilinear Relations of the Matrix Hierarchy Associated with a Commutative Algebra of
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An Algebraic Characterization of the Bilinear Relations of the Matrix Hierarchy Associated with a Commutative Algebra of k × k-Matrices Gerardus F. Helminck · Elena A. Panasenko
Received: 23 October 2008 / Accepted: 21 November 2008 / Published online: 10 February 2009 © Springer Science+Business Media B.V. 2009
Abstract In this paper we give a purely algebraic set-up for the equations of the matrix hierarchy that can be associated to a maximal commutative subalgebra of the k × k-matrices. Besides that it gives you a proper framework for the description of the linearization and the Lax form of the hierarchy, it enables you also to give an algebraic characterization of the dual wavefunctions of the matrix hierarchy and this leads to an algebraic interpretation of the bilinear form of this system of nonlinear equations. Keywords Matrix hierarchy · Linearization · Wavefunction · Dual wavefunction · Bilinear relations Mathematics Subject Classification (2000) 22E65 · 22E70 · 35Q58 · 58B25
1 Introduction Hierachies of nonlinear differential and difference equations were implicit in the work of many people, such as Gelfand, Dickey [8], Krichever, van Moerbeke, Wilson, and made their first explicit appearance in the work of the Sato school, see e.g. [3–5] and [15]. A popular formulation was the Lax form consisting of a tower of compatibility equations for the evolution of the coefficients of certain operators. The form of these operators varied from case to case. They could be scalar differential operators, like the Schrödinger operator in the case of the Korteweg-de Vries-hierarchy. The basic operators could also be pseudo-differential operators, a mixture of differential operators and integral operators like it was the case with
Supported by NWO-RFBR Grant 047.017.015. G.F. Helminck () KdV Institute, University of Amsterdam, Amsterdam, The Netherlands e-mail: [email protected] E.A. Panasenko Derzhavin Tambov State University, Internatsionalnaya 33, 392000 Tambov, Russia e-mail: [email protected]
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G.F. Helminck, E.A. Panasenko
the Kadomtsev-Petviashvili-hierarchy, and infinite matrices, which are the central objects in the Toda-hierarchy, see [15] and [1]. In the case of the N -wave equation, see [6], one dealt with matrix differential operators and in the multicomponent KP-hierarchy case, the relevant operators were matrix pseudo-differential operators. To give an idea, we recall the Lax-form of the KP-hierarchy, which is closest to what we will consider here. The evolution of the scalar pseudodifferential operator L in that case is described by infinitely many degrees of freedom {x, t1 , . . . , tn , . . .}. The coefficients of L belong to a ring R of functions in the parameters {x, tn } that is stable w.r.t. the privileged ∂ and all the derivations {∂n := ∂t∂n , n ≥ 1}. The operator L is a deformation derivation ∂ = ∂x of the operator ∂ by an integral operator and has the form L=∂ + lj ∂ j , with lj ∈ R. (1) j 0
∂˜iα (U˜ β ) := ∂˜iα (Eβ ) +
˜ −j = [((L) ˜ i U˜ α )+ , U˜ β ]. ∂˜iα (u(0) β,j )∂
(9)
j
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