64 research outputs found
Tau-Functions generating the Conservation Laws for Generalized Integrable Hierarchies of KdV and Affine-Toda type
For a class of generalized integrable hierarchies associated with affine
(twisted or untwisted) Kac-Moody algebras, an explicit representation of their
local conserved densities by means of a single scalar tau-function is deduced.
This tau-function acts as a partition function for the conserved densities,
which fits its potential interpretation as the effective action of some quantum
system. The class consists of multi-component generalizations of the
Drinfel'd-Sokolov and the two-dimensional affine Toda lattice hierarchies. The
relationship between the former and the approach of Feigin, Frenkel and
Enriquez to soliton equations of KdV and mKdV type is also discussed. These
results considerably simplify the calculation of the conserved charges carried
by the soliton solutions to the equations of the hierarchy, which is important
to establish their interpretation as particles. By way of illustration, we
calculate the charges carried by a set of constrained KP solitons recently
constructed.Comment: 47 pages, plain TeX with AMS fonts, no figure
Solitons, Tau-functions and Hamiltonian Reduction for Non-Abelian Conformal Affine Toda Theories
We consider the Hamiltonian reduction of the two-loop
Wess-Zumino-Novikov-Witten model (WZNW) based on an untwisted affine Kac-Moody
algebra \cgh. The resulting reduced models, called {\em Generalized
Non-Abelian Conformal Affine Toda (G-CAT)}, are conformally invariant and a
wide class of them possesses soliton solutions; these models constitute
non-abelian generalizations of the Conformal Affine Toda models. Their general
solution is constructed by the Leznov-Saveliev method. Moreover, the dressing
transformations leading to the solutions in the orbit of the vacuum are
considered in detail, as well as the -functions, which are defined for
any integrable highest weight representation of \cgh, irrespectively of its
particular realization. When the conformal symmetry is spontaneously broken,
the G-CAT model becomes a generalized Affine Toda model, whose soliton
solutions are constructed. Their masses are obtained exploring the spontaneous
breakdown of the conformal symmetry, and their relation to the fundamental
particle masses is discussed.Comment: 47 pages. LaTe
Confinement and soliton solutions in the SL(3) Toda model coupled to matter fields
We consider an integrable conformally invariant two dimensional model
associated to the affine Kac-Moody algebra SL(3). It possesses four scalar
fields and six Dirac spinors. The theory does not possesses a local Lagrangian
since the spinor equations of motion present interaction terms which are
bilinear in the spinors. There exists a submodel presenting an equivalence
between a U(1) vector current and a topological current, which leads to a
confinement of the spinors inside the solitons. We calculate the one-soliton
and two-soliton solutions using a procedure which is a hybrid of the dressing
and Hirota methods. The soliton masses and time delays due to the soliton
interactions are also calculated. We give a computer program to calculate the
soliton solutions.Comment: plain LaTeX, 37 page
Orthogonal Decomposition of Some Affine Lie Algebras in Terms of their Heisenberg Subalgebras
In the present note we suggest an affinization of a theorem by Kostrikin
et.al. about the decomposition of some complex simple Lie algebras
into the algebraic sum of pairwise orthogonal Cartan subalgebras. We point out
that the untwisted affine Kac-Moody algebras of types ( prime,
), can be decomposed into
the algebraic sum of pairwise or\-tho\-go\-nal Heisenberg subalgebras. The
and cases are discussed in great detail. Some possible
applications of such decompositions are also discussed.Comment: 16 pages, LaTeX, no figure
A New Approach to Integrable Theories in any Dimension
The zero curvature representation for two dimensional integrable models is
generalized to spacetimes of dimension d+1 by the introduction of a d-form
connection. The new generalized zero curvature conditions can be used to
represent the equations of motion of some relativistic invariant field theories
of physical interest in 2+1 dimensions (BF theories, Chern-Simons, 2+1 gravity
and the CP^1 model) and 3+1 dimensions (self-dual Yang-Mills theory and the
Bogomolny equations). Our approach leads to new methods of constructing
conserved currents and solutions. In a submodel of the 2+1 dimensional CP^1
model, we explicitly construct an infinite number of previously unknown
nontrivial conserved currents. For each positive integer spin representation of
sl(2) we construct 2j+1 conserved currents leading to 2j+1 Lorentz scalar
charges.Comment: 52 pages, 4 figures, shortened version to appear in NP
Solitons from Dressing in an Algebraic Approach to the Constrained KP Hierarchy
The algebraic matrix hierarchy approach based on affine Lie algebras
leads to a variety of 1+1 soliton equations. By varying the rank of the
underlying algebra as well as its gradation in the affine setting, one
encompasses the set of the soliton equations of the constrained KP hierarchy.
The soliton solutions are then obtained as elements of the orbits of the
dressing transformations constructed in terms of representations of the vertex
operators of the affine algebras realized in the unconventional
gradations. Such soliton solutions exhibit non-trivial dependence on the KdV
(odd) time flows and KP (odd and even) time flows which distinguishes them from
the conventional structure of the Darboux-B\"{a}cklund Wronskian solutions of
the constrained KP hierarchy.Comment: LaTeX, 13pg
Necessary Optimality Conditions for Higher-Order Infinite Horizon Variational Problems on Time Scales
We obtain Euler-Lagrange and transversality optimality conditions for
higher-order infinite horizon variational problems on a time scale. The new
necessary optimality conditions improve the classical results both in the
continuous and discrete settings: our results seem new and interesting even in
the particular cases when the time scale is the set of real numbers or the set
of integers.Comment: This is a preprint of a paper whose final and definite form will
appear in Journal of Optimization Theory and Applications (JOTA). Paper
submitted 17-Nov-2011; revised 24-March-2012 and 10-April-2012; accepted for
publication 15-April-201
Euler-Lagrange equations for composition functionals in calculus of variations on time scales
In this paper we consider the problem of the calculus of variations for a
functional which is the composition of a certain scalar function with the
delta integral of a vector valued field , i.e., of the form
. Euler-Lagrange
equations, natural boundary conditions for such problems as well as a necessary
optimality condition for isoperimetric problems, on a general time scale, are
given. A number of corollaries are obtained, and several examples illustrating
the new results are discussed in detail.Comment: Submitted 10-May-2009 to Discrete and Continuous Dynamical Systems
(DCDS-B); revised 10-March-2010; accepted 04-July-201
Quantum Supersymmetric Toda-mKdV Hierarchies
In this paper we generalize the quantization procedure of Toda-mKdV
hierarchies to the case of arbitrary affine (super)algebras. The quantum
analogue of the monodromy matrix, related to the universal R-matrix with the
lower Borel subalgebra represented by the corresponding vertex operators is
introduced. The auxiliary L-operators satisfying RTT-relation are constructed
and the quantum integrability condition is obtained. General approach is
illustrated by means of two important examples.Comment: LaTeX2e, elsart.cls, 21 pages, Nuclear Physics B, 2005, in pres
Massive symmetric space sine-Gordon soliton theories and perturbed conformal field theory
The perturbed conformal field theories corresponding to the massive Symmetric Space sine-Gordon soliton theories are identified by calculating the central charge of the unperturbed conformal field theory and the conformal dimension of the perturbation. They are described by an action with a positive-definite kinetic term and a real potential term bounded from below, their equations of motion are non-abelian affine Toda equations and, moreover, they exhibit a mass gap. The unperturbed CFT corresponding to the compact symmetric space G/G_0 is either the WZNW action for G_0 or the gauged WZNW action for a coset of the form G_0/U(1)^p. The quantum integrability of the theories that describe perturbations of a WZNW action, named Split models, is established by showing that they have quantum conserved quantities of spin +3 and -3. Together with the already known results for the other massive theories associated with the non-abelian affine Toda equations, the Homogeneous sine-Gordon theories, this supports the conjecture that all the massive Symmetric Space sine-Gordon theories will be quantum integrable and, hence, will admit a factorizable S-matrix. The general features of the soliton spectrum are discussed, and some explicit soliton solutions for the Split models are constructed. In general, the solitons will carry both topological charges and abelian Noether charges. Moreover, the spectrum is expected to include stable and unstable particles
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