592 research outputs found
Non-degenerate solutions of universal Whitham hierarchy
The notion of non-degenerate solutions for the dispersionless Toda hierarchy
is generalized to the universal Whitham hierarchy of genus zero with
marked points. These solutions are characterized by a Riemann-Hilbert problem
(generalized string equations) with respect to two-dimensional canonical
transformations, and may be thought of as a kind of general solutions of the
hierarchy. The Riemann-Hilbert problem contains arbitrary functions
, , which play the role of generating functions of
two-dimensional canonical transformations. The solution of the Riemann-Hilbert
problem is described by period maps on the space of -tuples
of conformal maps from disks of the
Riemann sphere and their complements to the Riemann sphere. The period maps are
defined by an infinite number of contour integrals that generalize the notion
of harmonic moments. The -function (free energy) of these solutions is also
shown to have a contour integral representation.Comment: latex2e, using amsmath, amssym and amsthm packages, 32 pages, no
figur
From finite geometry exact quantities to (elliptic) scattering amplitudes for spin chains: the 1/2-XYZ
Initially, we derive a nonlinear integral equation for the vacuum counting
function of the spin 1/2-XYZ chain in the {\it disordered regime}, thus
paralleling similar results by Kl\"umper \cite{KLU}, achieved through a
different technique in the {\it antiferroelectric regime}. In terms of the
counting function we obtain the usual physical quantities, like the energy and
the transfer matrix (eigenvalues). Then, we introduce a double scaling limit
which appears to describe the sine-Gordon theory on cylindrical geometry, so
generalising famous results in the plane by Luther \cite{LUT} and Johnson et
al. \cite{JKM}. Furthermore, after extending the nonlinear integral equation to
excitations, we derive scattering amplitudes involving solitons/antisolitons
first, and bound states later. The latter case comes out as manifestly related
to the Deformed Virasoro Algebra of Shiraishi et al. \cite{SKAO}. Although this
nonlinear integral equations framework was contrived to deal with finite
geometries, we prove it to be effective for discovering or rediscovering
S-matrices. As a particular example, we prove that this unique model furnishes
explicitly two S-matrices, proposed respectively by Zamolodchikov \cite{ZAMe}
and Lukyanov-Mussardo-Penati \cite{LUK, MP} as plausible scattering description
of unknown integrable field theories.Comment: Article, 41 pages, Late
Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking
We have recently solved the inverse spectral problem for integrable PDEs in
arbitrary dimensions arising as commutation of multidimensional vector fields
depending on a spectral parameter . The associated inverse problem, in
particular, can be formulated as a non linear Riemann Hilbert (NRH) problem on
a given contour of the complex plane. The most distinguished examples
of integrable PDEs of this type, like the dispersionless
Kadomtsev-Petviashivili (dKP), the heavenly and the 2 dimensional
dispersionless Toda equations, are real PDEs associated with Hamiltonian vector
fields. The corresponding NRH data satisfy suitable reality and symplectic
constraints. In this paper, generalizing the examples of solvable NRH problems
illustrated in \cite{MS4,MS5,MS6}, we present a general procedure to construct
solvable NRH problems for integrable real PDEs associated with Hamiltonian
vector fields, allowing one to construct implicit solutions of such PDEs
parametrized by an arbitrary number of real functions of a single variable.
Then we illustrate this theory on few distinguished examples for the dKP and
heavenly equations. For the dKP case, we characterize a class of similarity
solutions, a class of solutions constant on their parabolic wave front and
breaking simultaneously on it, and a class of localized solutions breaking in a
point of the plane. For the heavenly equation, we characterize two
classes of symmetry reductions.Comment: 29 page
SDiff(2) Toda equation -- hierarchy, function, and symmetries
A continuum limit of the Toda lattice field theory, called the SDiff(2) Toda
equation, is shown to have a Lax formalism and an infinite hierarchy of higher
flows. The Lax formalism is very similar to the case of the self-dual vacuum
Einstein equation and its hyper-K\"ahler version, however now based upon a
symplectic structure and the group SDiff(2) of area preserving diffeomorphisms
on a cylinder . An analogue of the Toda lattice tau function is
introduced. The existence of hidden SDiff(2) symmetries are derived from a
Riemann-Hilbert problem in the SDiff(2) group. Symmetries of the tau function
turn out to have commutator anomalies, hence give a representation of a central
extension of the SDiff(2) algebra.Comment: 16 pages (``vanilla.sty" is attatched to the end of this file after
``\bye" command
-analogue of modified KP hierarchy and its quasi-classical limit
A -analogue of the tau function of the modified KP hierarchy is defined by
a change of independent variables. This tau function satisfies a system of
bilinear -difference equations. These bilinear equations are translated to
the language of wave functions, which turn out to satisfy a system of linear
-difference equations. These linear -difference equations are used to
formulate the Lax formalism and the description of quasi-classical limit. These
results can be generalized to a -analogue of the Toda hierarchy. The results
on the -analogue of the Toda hierarchy might have an application to the
random partition calculus in gauge theories and topological strings.Comment: latex2e, a4 paper 15 pages, no figure; (v2) a few references are
adde
Eigenvalues of Ruijsenaars-Schneider models associated with root system in Bethe ansatz formalism
Ruijsenaars-Schneider models associated with root system with a
discrete coupling constant are studied. The eigenvalues of the Hamiltonian are
givein in terms of the Bethe ansatz formulas. Taking the "non-relativistic"
limit, we obtain the spectrum of the corresponding Calogero-Moser systems in
the third formulas of Felder et al [20].Comment: Latex file, 25 page
Integrable Time-Discretisation of the Ruijsenaars-Schneider Model
An exactly integrable symplectic correspondence is derived which in a
continuum limit leads to the equations of motion of the relativistic
generalization of the Calogero-Moser system, that was introduced for the first
time by Ruijsenaars and Schneider. For the discrete-time model the equations of
motion take the form of Bethe Ansatz equations for the inhomogeneous spin-1/2
Heisenberg magnet. We present a Lax pair, the symplectic structure and prove
the involutivity of the invariants. Exact solutions are investigated in the
rational and hyperbolic (trigonometric) limits of the system that is given in
terms of elliptic functions. These solutions are connected with discrete
soliton equations. The results obtained allow us to consider the Bethe Ansatz
equations as ones giving an integrable symplectic correspondence mixing the
parameters of the quantum integrable system and the parameters of the
corresponding Bethe wavefunction.Comment: 27 pages, latex, equations.st
The multicomponent 2D Toda hierarchy: Discrete flows and string equations
The multicomponent 2D Toda hierarchy is analyzed through a factorization
problem associated to an infinite-dimensional group. A new set of discrete
flows is considered and the corresponding Lax and Zakharov--Shabat equations
are characterized. Reductions of block Toeplitz and Hankel bi-infinite matrix
types are proposed and studied. Orlov--Schulman operators, string equations and
additional symmetries (discrete and continuous) are considered. The
continuous-discrete Lax equations are shown to be equivalent to a factorization
problem as well as to a set of string equations. A congruence method to derive
site independent equations is presented and used to derive equations in the
discrete multicomponent KP sector (and also for its modification) of the theory
as well as dispersive Whitham equations.Comment: 27 pages. In the revised paper we improved the presentatio
Loewner equations, Hirota equations and reductions of universal Whitham hierarchy
This paper reconsiders finite variable reductions of the universal Whitham
hierarchy of genus zero in the perspective of dispersionless Hirota equations.
In the case of one-variable reduction, dispersionless Hirota equations turn out
to be a powerful tool for understanding the mechanism of reduction. All
relevant equations describing the reduction (L\"owner-type equations and
diagonal hydrodynamic equations) can be thereby derived and justified in a
unified manner. The case of multi-variable reductions is not so
straightforward. Nevertheless, the reduction procedure can be formulated in a
general form, and justified with the aid of dispersionless Hirota equations. As
an application, previous results of Guil, Ma\~{n}as and Mart\'{\i}nez Alonso
are reconfirmed in this formulation.Comment: latex 2e using packages amsmath,amssymb,amsthm, 39 pages, no figure;
(v2) a few typos corrected and accepted for publicatio
Cathodoluminescence characterization of Ge-doped CdTe crystals
Cathodoluminescence (CL) microscopic techniques have been used to study the spatial distribution of structural defects and the deep levels in CdTe:Ge bulk crystals. The effect of Ge doping with concentrations of 10(17) and 10(19) cm(-3) on the compensation of V-Cd in CdTe has been investigated. Dependence of the intensity distribution of CL emission bands on the dopant concentration has been studied. Ge doping causes a substantial reduction of the generally referred to 1.40 eV luminescence, which is often present in undoped CdTe crystals, and enhances the 0.91 and 0.81 eV emissions
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