638 research outputs found
Differential Calculi on Associative Algebras and Integrable Systems
After an introduction to some aspects of bidifferential calculus on
associative algebras, we focus on the notion of a "symmetry" of a generalized
zero curvature equation and derive Backlund and (forward, backward and binary)
Darboux transformations from it. We also recall a matrix version of the binary
Darboux transformation and, inspired by the so-called Cauchy matrix approach,
present an infinite system of equations solved by it. Finally, we sketch recent
work on a deformation of the matrix binary Darboux transformation in
bidifferential calculus, leading to a treatment of integrable equations with
sources.Comment: 19 pages, to appear in "Algebraic Structures and Applications", S.
Silvestrov et al (eds.), Springer Proceedings in Mathematics & Statistics,
202
Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy
New extensions of the KP and modified KP hierarchies with self-consistent
sources are proposed. The latter provide new generalizations of
-dimensional integrable equations, including the DS-III equation and the
-wave problem. Furthermore, we recover a system that contains two types of
the KP equation with self-consistent sources as special cases. Darboux and
binary Darboux transformations are applied to generate solutions of the
proposed hierarchies
On Integrability and Exact Solvability in Deterministic and Stochastic Laplacian Growth
We review applications of theory of classical and quantum integrable systems
to the free-boundary problems of fluid mechanics as well as to corresponding
problems of statistical mechanics. We also review important exact results
obtained in the theory of multi-fractal spectra of the stochastic models
related to the Laplacian growth: Schramm-Loewner and Levy-Loewner evolutions
The q-deformed mKP hierarchy with self-consistent sources, Wronskian solutions and solitons
Based on the eigenfunction symmetry constraint of the q-deformed modified KP hierarchy, a q-deformed mKP hierarchy with self-consistent sources (q-mKPHSCSs) is constructed. The q-mKPHSCSs contain two types of q-deformed mKP equation with self-consistent sources. By the combination of the dressing method and the method of variation of constants, a generalized dressing approach is proposed to solve the q-deformed KP hierarchy with self-consistent sources (q-KPHSCSs). Using the gauge transformation between the q-KPHSCSs and the q-mKPHSCSs, the q-deformed Wronskian solutions for the q-KPHSCSs and the q-mKPHSCSs are obtained. The one-soliton solutions for the q-deformed KP (mKP) equation with a source are given explicitly
The Lax Integrable Differential-Difference Dynamical Systems on Extended Phase Spaces
The Hamiltonian representation for the hierarchy of Lax-type flows on a dual
space to the Lie algebra of shift operators coupled with suitable
eigenfunctions and adjoint eigenfunctions evolutions of associated spectral
problems is found by means of a specially constructed Backlund transformation.
The Hamiltonian description for the corresponding set of squared eigenfunction
symmetry hierarchies is represented. The relation of these hierarchies with Lax
integrable (2+1)-dimensional differential-difference systems and their triple
Lax-type linearizations is analysed. The existence problem of a Hamiltonian
representation for the coupled Lax-type hierarchy on a dual space to the
central extension of the shift operator Lie algebra is solved also
Discretisations of constrained KP hierarchies
We present a discrete analogue of the so-called symmetry reduced or
`constrained' KP hierarchy. As a result we obtain integrable discretisations,
in both space and time, of some well-known continuous integrable systems such
as the nonlinear Schroedinger equation, the Broer-Kaup equation and the
Yajima-Oikawa system, together with their Lax pairs. It will be shown that
these discretisations also give rise to a discrete description of the entire
hierarchy of associated integrable systems. The discretisations of the
Broer-Kaup equation and of the Yajima-Oikawa system are thought to be new.Comment: Accepted for publication in Journal of Mathematical Sciences, The
University of Toky
Covariant Poisson equation with compact Lie algebras
The covariant Poisson equation for Lie algebra-valued mappings defined in
3-dimensional Euclidean space is studied using functional analytic methods.
Weighted covariant Sobolev spaces are defined and used to derive sufficient
conditions for the existence and smoothness of solutions to the covariant
Poisson equation. These conditions require, apart from suitable continuity,
appropriate local integrability of the gauge potentials and global weighted
integrability of the curvature form and the source. The possibility of
nontrivial asymptotic behaviour of a solution is also considered. As a
by-product, weighted covariant generalisations of Sobolev embeddings are
established.Comment: 31 pages, LaTeX2
Flux compactifications in string theory: a comprehensive review
We present a pedagogical overview of flux compactifications in string theory,
from the basic ideas to the most recent developments. We concentrate on closed
string fluxes in type II theories. We start by reviewing the supersymmetric
flux configurations with maximally symmetric four-dimensional spaces. We then
discuss the no-go theorems (and their evasion) for compactifications with
fluxes. We analyze the resulting four-dimensional effective theories, as well
as some of its perturbative and non-perturbative corrections, focusing on
moduli stabilization. Finally, we briefly review statistical studies of flux
backgrounds.Comment: 85 pages, 2 figures. v2, v3: minor changes, references adde
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