67 research outputs found
The vector k-constrained KP hierarchy and Sato's Grassmannian
We use the representation theory of the infinite matrix group to show that
(in the polynomial case) the --vector --constrained KP hierarchy has a
natural geometrical interpretation on Sato's infinite Grassmannian. This
description generalizes the the --reduced KP or Gelfand--Dickey hierarchies.Comment: 15 pages, AMSTe
The Adler-Shiota-van Moerbeke formula for the BKP hierarchy
We study the BKP hierarchy and prove the existence of an Adler--Shiota--van
Moerbeke formula. This formula relates the action of the
--algebra on tau--functions to the action of the ``additional
symmetries'' on wave functions.Comment: 11 pages of plain tex, no figure
The (n,1)-Reduced DKP Hierarchy, the String Equation and W Constraints
The total descendent potential of a simple singularity satisfies the
Kac-Wakimoto principal hierarchy. Bakalov and Milanov showed recently that it
is also a highest weight vector for the corresponding W-algebra. This was used
by Liu, Yang and Zhang to prove its uniqueness. We construct this principal
hierarchy of type D in a different way, viz. as a reduction of some DKP
hierarchy. This gives a Lax type and a Grassmannian formulation of this
hierarchy. We show in particular that the string equation induces a large part
of the W constraints of Bakalov and Milanov. These constraints are not only
given on the tau function, but also in terms of the Lax and Orlov-Schulman
operators
Clifford Algebra Derivations of Tau-Functions for Two-Dimensional Integrable Models with Positive and Negative Flows
We use a Grassmannian framework to define multi-component tau functions as
expectation values of certain multi-component Fermi operators satisfying simple
bilinear commutation relations on Clifford algebra. The tau functions contain
both positive and negative flows and are shown to satisfy the -component KP
hierarchy. The hierarchy equations can be formulated in terms of
pseudo-differential equations for matrix wave functions derived in
terms of tau functions. These equations are cast in form of Sato-Wilson
relations. A reduction process leads to the AKNS, two-component Camassa-Holm
and Cecotti-Vafa models and the formalism provides simple formulas for their
solutionsComment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
The geometry of spinors and the multicomponent BKP and DKP hierarchies
We develop a formalism of multicomponent BKP hierarchies using elementary
geometry of spinors. The multicomponent KP and the modified KP hierarchy (hence
all their reductions like KdV, NLS, AKNS or DS) are reductions of the
multicomponent BKP.Comment: 46 pages, Latex2
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