40,747 research outputs found
Tau-functions and Dressing Transformations for Zero-Curvature Affine Integrable Equations
The solutions of a large class of hierarchies of zero-curvature equations
that includes Toda and KdV type hierarchies are investigated. All these
hierarchies are constructed from affine (twisted or untwisted) Kac-Moody
algebras~. Their common feature is that they have some special ``vacuum
solutions'' corresponding to Lax operators lying in some abelian (up to the
central term) subalgebra of~; in some interesting cases such subalgebras
are of the Heisenberg type. Using the dressing transformation method, the
solutions in the orbit of those vacuum solutions are constructed in a uniform
way. Then, the generalized tau-functions for those hierarchies are defined as
an alternative set of variables corresponding to certain matrix elements
evaluated in the integrable highest-weight representations of~. Such
definition of tau-functions applies for any level of the representation, and it
is independent of its realization (vertex operator or not). The particular
important cases of generalized mKdV and KdV hierarchies as well as the abelian
and non abelian affine Toda theories are discussed in detail.Comment: 27 pages, plain Te
A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity
We consider a one dimensional transport model with nonlocal velocity given by
the Hilbert transform and develop a global well-posedness theory of probability
measure solutions. Both the viscous and non-viscous cases are analyzed. Both in
original and in self-similar variables, we express the corresponding equations
as gradient flows with respect to a free energy functional including a singular
logarithmic interaction potential. Existence, uniqueness, self-similar
asymptotic behavior and inviscid limit of solutions are obtained in the space
of probability measures with finite second
moments, without any smallness condition. Our results are based on the abstract
gradient flow theory developed in \cite{Ambrosio}. An important byproduct of
our results is that there is a unique, up to invariance and translations,
global in time self-similar solution with initial data in
, which was already obtained in
\textrm{\cite{Deslippe,Biler-Karch}} by different methods. Moreover, this
self-similar solution attracts all the dynamics in self-similar variables. The
crucial monotonicity property of the transport between measures in one
dimension allows to show that the singular logarithmic potential energy is
displacement convex. We also extend the results to gradient flow equations with
negative power-law locally integrable interaction potentials
The role of the RM-ODP computational viewpoint concepts in the MDA approach
An MDA design approach should be able to accommodate designs at different levels of platform-independence. We have proposed a design approach previously (in [2]), which allows these levels to be identified. An important feature of this approach is the notion of abstract platform. An abstract platform is determined by the platform characteristics that are relevant for applications at a certain level of platform-independence, and must be established by considering various design goals. In this paper, we define a framework that makes it possible to use RM-ODP concepts in our MDA design approach. This framework allows a recursive application of the computational viewpoint at different levels of platform-independence. This is obtained by equating the RM-ODP notion of infrastructure to our notion of abstract platform
The structures underlying soliton solutions in integrable hierarchies
We point out that a common feature of integrable hierarchies presenting
soliton solutions is the existence of some special ``vacuum solutions'' such
that the Lax operators evaluated on them, lie in some abelian subalgebra of the
associated Kac-Moody algebra. The soliton solutions are constructed out of
those ``vacuum solitons'' by the dressing transformation procedure.Comment: Talk given at the I Latin American Symposium on High Energy Physics,
I SILAFAE, Merida, Mexico, November/96, 5 pages, LaTeX, needs aipproc.tex,
aipproc.sty, aipproc.cls, available from
ftp://ftp.aip.org/ems/tex/macros/proceedings/6x9
Some Comments on BPS systems
We look at simple BPS systems involving more than one field. We discuss the
conditions that have to be imposed on various terms in Lagrangians involving
many fields to produce BPS systems and then look in more detail at the simplest
of such cases. We analyse in detail BPS systems involving 2 interacting
Sine-Gordon like fields, both when one of them has a kink solution and the
second one either a kink or an antikink solution. We take their solitonic
static solutions and use them as initial conditions for their evolution in
Lorentz covariant versions of such models. We send these structures towards
themselves and find that when they interact weakly they can pass through each
other with a phase shift which is related to the strength of their interaction.
When they interact strongly they repel and reflect on each other. We use the
method of a modified gradient flow in order to visualize the solutions in the
space of fields.Comment: 27 pages, 17 figure
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