307,643 research outputs found
On a Lagrangian reduction and a deformation of completely integrable systems
We develop a theory of Lagrangian reduction on loop groups for completely
integrable systems after having exchanged the role of the space and time
variables in the multi-time interpretation of integrable hierarchies. We then
insert the Sobolev norm in the Lagrangian and derive a deformation of the
corresponding hierarchies. The integrability of the deformed equations is
altered and a notion of weak integrability is introduced. We implement this
scheme in the AKNS and SO(3) hierarchies and obtain known and new equations.
Among them we found two important equations, the Camassa-Holm equation, viewed
as a deformation of the KdV equation, and a deformation of the NLS equation
Commuting Flows and Conservation Laws for Noncommutative Lax Hierarchies
We discuss commuting flows and conservation laws for Lax hierarchies on
noncommutative spaces in the framework of the Sato theory. On commutative
spaces, the Sato theory has revealed essential aspects of the integrability for
wide class of soliton equations which are derived from the Lax hierarchies in
terms of pseudo-differential operators. Noncommutative extension of the Sato
theory has been already studied by the author and Kouichi Toda, and the
existence of various noncommutative Lax hierarchies are guaranteed. In the
present paper, we present conservation laws for the noncommutative Lax
hierarchies with both space-space and space-time noncommutativities and prove
the existence of infinite number of conserved densities. We also give the
explicit representations of them in terms of Lax operators. Our results include
noncommutative versions of KP, KdV, Boussinesq, coupled KdV, Sawada-Kotera,
modified KdV equations and so on.Comment: 22 pages, LaTeX, v2: typos corrected, references added, version to
appear in JM
From Additional Symmetries to Linearization of Virasoro Symmetries
We construct the additional symmetries and derive the Adler-Shiota-van
Moerbeke formula for the two-component BKP hierarchy. We also show that the
Drinfeld-Sokolov hierarchies of type D, which are reduced from the
two-component BKP hierarchy, possess symmetries written as the action of a
series of linear Virasoro operators on the tau function. It results in that the
Drinfeld-Sokolov hierarchies of type D coincide with Dubrovin and Zhang's
hierarchies associated to the Frobenius manifolds for Coxeter groups of type D,
and that every solution of such a hierarchy together with the string equation
is annihilated by certain combinations of the Virasoro operators and the time
derivations of the hierarchy.Comment: 22 page
Non-Abelian coset string backgrounds from asymptotic and initial data
We describe hierarchies of exact string backgrounds obtained as non-Abelian
cosets of orthogonal groups and having a space--time realization in terms of
gauged WZW models. For each member in these hierarchies, the target-space
backgrounds are generated by the ``boundary'' backgrounds of the next member.
We explicitly demonstrate that this property holds to all orders in .
It is a consequence of the existence of an integrable marginal operator build
on, generically, non-Abelian parafermion bilinears. These are dressed with the
dilaton supported by the extra radial dimension, whose asymptotic value defines
the boundary. Depending on the hierarchy, this boundary can be time-like or
space-like with, in the latter case, potential cosmological applications.Comment: 26 page
Grassmannian Approach to Super KP Hierarchies
We present a theory of 'maximal' super-KP(SKP) hierarchy whose flows are
maximally extended to include all those of known SKP hierarchies, including,
for example, the MRSKP hierarchy of Manin and Radul and the Jacobian SKP(JSKP)
introduced by Mulase and Rabin. It is shown that SKP hierarchies has a natural
field theoretic description in terms of the B-C system, in analogous way as the
ordinary KP hierarchy. For this SKP hierarchy, we construct the vertex
operators by using Kac-van de Leur superbosonization. The vertex operators act
on the -function and then produce the wave function and the dual wave
function of the hierarchy. Thereby we achieve the description of the 'maximal'
SKP hierarchy in terms of the -function, which seemed to be lacking
till now. Mutual relations among the SKP hierarchies are clarified. The MRSKP
and the JSKP hierarchies are obtained as special cases when the time variables
are appropriately restricted.Comment: 46 pages, LaTex, no figure ( few typos corrected
Cache Hierarchy Inspired Compression: a Novel Architecture for Data Streams
We present an architecture for data streams based on structures typically found in web cache hierarchies. The main idea is to build a meta level analyser from a number of levels constructed over time from a data stream. We present the general architecture for such a system and an application to classification. This architecture is an instance of the general wrapper idea allowing us to reuse standard batch learning algorithms in an inherently incremental learning environment. By artificially generating data sources we demonstrate that a hierarchy containing a mixture of models is able to adapt over time to the source of the data. In these experiments the hierarchies use an elementary performance based replacement policy and unweighted voting for making classification decisions
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