1,836 research outputs found
Detecting and determining preserved measures and integrals of rational maps
In this paper we use the method of discrete Darboux polynomials to calculate
preserved measures and integrals of rational maps. The approach is based on the
use of cofactors and Darboux polynomials and relies on the use of symbolic
algebra tools. Given sufficient computing power, most, if not all, rational
preserved integrals can be found (and even some non-rational ones).
We show, in a number of examples, how it is possible to use this method to
both determine and detect preserved measures and integrals of the considered
rational maps. Many of the examples arise from the Kahan-Hirota-Kimura
discretization of completely integrable systems of ordinary differential
equations
Geometry of Darboux-Manakov-Zakharov systems and its application
The intrinsic geometric properties of generalized Darboux-Manakov-Zakharov
systems of semilinear partial differential equations \label{GDMZabstract}
\frac{\partial^2 u}{\partial x_i\partial x_j}=f_{ij}\Big(x_k,u,\frac{\partial
u}{\partial x_l}\Big), 1\leq i<j\leq n, k,l\in\{1,...,n\} for a real-valued
function are studied with particular reference to the linear
systems in this equation class.
System (\ref{GDMZabstract}) will not generally be involutive in the sense of
Cartan: its coefficients will be constrained by complicated nonlinear
integrability conditions. We derive geometric tools for explicitly constructing
involutive systems of the form (\ref{GDMZabstract}), essentially solving the
integrability conditions. Specializing to the linear case provides us with a
novel way of viewing and solving the multi-dimensional -wave resonant
interaction system and its modified version as well as constructing new
examples of semi-Hamiltonian systems of hydrodynamic type. The general theory
is illustrated by a study of these applications
Quantum Super-Integrable Systems as Exactly Solvable Models
We consider some examples of quantum super-integrable systems and the
associated nonlinear extensions of Lie algebras. The intimate relationship
between super-integrability and exact solvability is illustrated.
Eigenfunctions are constructed through the action of the commuting operators.
Finite dimensional representations of the quadratic algebras are thus
constructed in a way analogous to that of the highest weight representations of
Lie algebras.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Twistor Approach to String Compactifications: a Review
We review a progress in obtaining the complete non-perturbative effective
action of type II string theory compactified on a Calabi-Yau manifold. This
problem is equivalent to understanding quantum corrections to the metric on the
hypermultiplet moduli space. We show how all these corrections, which include
D-brane and NS5-brane instantons, are incorporated in the framework of the
twistor approach, which provides a powerful mathematical description of
hyperkahler and quaternion-Kahler manifolds. We also present new insights on
S-duality, quantum mirror symmetry, connections to integrable models and
topological strings.Comment: 99 pages; minor corrections; journal versio
Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations
We present a general solution-generating result within the bidifferential
calculus approach to integrable partial differential and difference equations,
based on a binary Darboux-type transformation. This is then applied to the
non-autonomous chiral model, a certain reduction of which is known to appear in
the case of the D-dimensional vacuum Einstein equations with D-2 commuting
Killing vector fields. A large class of exact solutions is obtained, and the
aforementioned reduction is implemented. This results in an alternative to the
well-known Belinski-Zakharov formalism. We recover relevant examples of
space-times in dimensions four (Kerr-NUT, Tomimatsu-Sato) and five (single and
double Myers-Perry black holes, black saturn, bicycling black rings)
Eisenhart Lift of --Dimensional Mechanics
The Eisenhart lift is a variant of geometrization of classical mechanics with
degrees of freedom in which the equations of motion are embedded into the
geodesic equations of a Brinkmann-type metric defined on -dimensional
spacetime of Lorentzian signature. In this work, the Eisenhart lift of
-dimensional mechanics on curved background is studied. The corresponding
-dimensional metric is governed by two scalar functions which are just the
conformal factor and the potential of the original dynamical system. We derive
a conformal symmetry and a corresponding quadratic integral, associated with
the Eisenhart lift. The energy--momentum tensor is constructed which, along
with the metric, provides a solution to the Einstein equations. Uplifts of
-dimensional superintegrable models are discussed with a particular emphasis
on the issue of hidden symmetries. It is shown that for the -dimensional
Darboux--Koenigs metrics, only type I can result in Eisenhart lifts which
satisfy the weak energy condition. However, some physically viable metrics with
hidden symmetries are presented.Comment: 20 page
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