An Analytic Description of the Vector Constrained KP Hierarchy

Helminck, G.F.; Leur, Johan van de

doi:10.1007/s002200050341

Cited by 24 publications

(23 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This means that given a τ function for KP with corresponding Lax pseudodifferential operator L we say that τ ∈ cKP m,n iff L m can be written as the ratio of two differential operators of order m + n and n respectively. These special reductions of KP begun to be studied in 1995 by Dickey and Krichever ([18], [19]); a geometric interpretation of corresponding points in the Grassmannian has been given in [21] and [22]. Going back to theorem 3.16 we point out that the first decomposition as well as the recursion formula are already known and, as pointed out in [20], come simply from the fact that we have a truncated dressing.…”

Section: Gl(n C)mentioning

confidence: 90%

Block Toeplitz Determinants, Constrained KP and Gelfand-Dickey Hierarchies

Cafasso

2008

Math Phys Anal Geom

View full text Add to dashboard Cite

We propose a method for computing any Gelfand-Dickey τ function living in SegalWilson Grassmannian as the asymptotics of block Toeplitz determinant associated to a certain class of symbols W(t; z). Also truncated block Toeplitz determinants associated to the same symbols are shown to be τ function for rational reductions of KP. Connection with Riemann-Hilbert problems is investigated both from the point of view of integrable systems and block Toeplitz operator theory. Examples of applications to algebro-geometric solutions are given. IntroductionThis paper deals with the applications of block Toeplitz determinants and their asymptotics to the study of integrable hierarchies. Asymptotics of block Toeplitz determinants and their applications to physics is a developing field of research; in recent years it has been shown how to compute some physically relevant quantities (e.g. correlation functions) studying asymptotics of some block Toeplitz determinants (see [25], [26], [27]). In particular in [25] and [26] authors, for the first time, showed effective computations for the case of block Toeplitz determinants with symbols that do not have half truncated Fourier series. This is of particular interest for us as, with our approach, we will be able to do the same for certain block Toeplitz determinants associated to algebro-geometric solutions of Gelfand Dickey hierarchies. Let us mention some theoretical results about (block) Toeplitz determinants we will use in this paper. Given a function γ(z) on the circle we denote T N (γ) the Toeplitz matrix with symbol γ given byWe use the term block Toeplitz for the case of matrix-valued symbol γ(z). In that case the entries γ (j−i) of the above matrix are n × n matrices themselves. We denoteand we use the notation T (γ) for the N × N matrix obtained letting N go to infinity. where G(γ) is a normalizing constant and the operator T (γ)T (γ −1 ) is such that its determinant is well defined as a Fredholm determinant (see section 2 for the precise statement). Once the asymptotics had been computed the next quite natural question was to find an expressions relating directly D N (γ), and not just its asymptotics, to certain Fredholm determinants. The problem was solved many years later by Borodin and Okounkov in [9] for the scalar case and generalized, in the same year, for matrix case by E. Basor and H.Widom in [10]. For matrix valued case Borodin-Okounkov formula reads(here we assume G(γ) = 1). The operator (I − K γ,N ) can be written explicitely in coordinates knowing certain Riemann-Hilbert factorizations of γ. Its Fredholm determinant is well defined (see section 2 for details). Now many proofs of Borodin-Okounkov formula are known (for instance [11] contains another proof of the same formula, see also the earlier paper [12]).In this paper we apply block-Toeplitz determinants to the computation of τ function of an (almost) arbitrary solution of Gelfand-Dickey hierarchy(L differential operator of order n, j = nk). More precisely to a given pointin the big cell of Segal-...

show abstract

Section: Gl(n C)mentioning

confidence: 90%

Block Toeplitz Determinants, Constrained KP and Gelfand-Dickey Hierarchies

Cafasso

2008

Math Phys Anal Geom

View full text Add to dashboard Cite

show abstract

“…The polynomial CKP Sato Grassmannian consists of linear subspaces of Gr (0) (H) such that for any f (z), g(z) ∈ W one has (f (z), g(−z)) = 0. To describe the spaces corresponding to the 2-vector 1-constrained CKP hierarchy, such W must also satisfy the following condition [11], [12], [3]. There exists a subspace (6.5) W ⊂ W of codimension 2 such that zW ⊂ W. We assume that there is no larger subspace W with zW ⊂ W .…”

Section: The Space Gr(h) Has a Subdivision Into Different Componentsmentioning

confidence: 99%

The symplectic Kadomtsev-Petviashvili hierarchy and rational solutions of Painlevé VI

Aratyn

Leur²

2005

Annales De L’institut Fourier

View full text Add to dashboard Cite

Equivalence is established between a special class of Painleve VI equations parametrized by a conformal dimension µ, time dependent Euler top equations, isomonodromic deformations and three-dimensional Frobenius manifolds. The isomodromic tau function and solutions of the Euler top equations are explicitly constructed in terms of Wronskian solutions of the 2-vector 1-constrained symplectic Kadomtsev-Petviashvili (CKP) hierarchy by means of Grassmannian formulation. These Wronskian solutions give rational solutions of the Painleve VI equation for µ = 1, 2, . . ..

show abstract

“…They often possess a geometric description in terms of the Grassmann manifold, see e.g. [5,6,9]. In this paper one considers inside the lower triangular Toda hierarchies analogues and generalizations of reductions like the nth Gelfand-Dickey hierarchy and the AKNS-hierarchy and presents here their geometric description.…”

mentioning

confidence: 98%

Reductions of Lower Triangular Toda Hierarchies

2007

View full text Add to dashboard Cite

Deforming commutative algebras in the lower triangular (Z × Z)-matrices yields lower triangular Toda hierarchies and their associated nonlinear equations. Like for their counterpart in the ring of pseudodifferential operators, the KP-hierarchy, one also has for these hierarchies a geometric picture: certain infinite chains of subspaces in an separable Hilbert space provide solutions of lower triangular Toda hierarchies. The KP-hierarchy and its multi-component version contain many interesting subsystems, like e.g. the nth GelfandDickey hierarchy and the AKNS-hierarchy. In this paper one considers analogues of these two subsystems in the context of the lower triangular Toda hierarchies and a geometric description of solutions to both type reductions is given.

show abstract

An Analytic Description of the Vector Constrained KP Hierarchy

Cited by 24 publications

References 25 publications

Block Toeplitz Determinants, Constrained KP and Gelfand-Dickey Hierarchies

Block Toeplitz Determinants, Constrained KP and Gelfand-Dickey Hierarchies

The symplectic Kadomtsev-Petviashvili hierarchy and rational solutions of Painlevé VI

Reductions of Lower Triangular Toda Hierarchies

Contact Info

Product

Resources

About