692 research outputs found
A Note on Fractional KdV Hierarchies
We introduce a hierarchy of mutually commuting dynamical systems on a finite
number of Laurent series. This hierarchy can be seen as a prolongation of the
KP hierarchy, or a ``reduction'' in which the space coordinate is identified
with an arbitrarily chosen time of a bigger dynamical system. Fractional KdV
hierarchies are gotten by means of further reductions, obtained by constraining
the Laurent series. The case of sl(3)^2 and its bihamiltonian structure are
discussed in detail.Comment: Final version to appear in J. Math. Phys. Some changes in the order
of presentation, with more emphasis on the geometrical picture. One figure
added (using epsf.sty). 30 pages, Late
A Novel Hierarchy of Integrable Lattices
In the framework of the reduction technique for Poisson-Nijenhuis structures,
we derive a new hierarchy of integrable lattice, whose continuum limit is the
AKNS hierarchy. In contrast with other differential-difference versions of the
AKNS system, our hierarchy is endowed with a canonical Poisson structure and,
moreover, it admits a vector generalisation. We also solve the associated
spectral problem and explicity contruct action-angle variables through the
r-matrix approach.Comment: Latex fil
Data-driven modeling for drop size distributions
The prediction of the drop size distribution (DSD) resulting from liquid atomization is key to the optimization of multiphase flows from gas-turbine propulsion through agriculture to healthcare. Obtaining high-fidelity data of liquid atomization, either experimentally or numerically, is expensive, which makes the exploration of the design space difficult. First, to tackle these challenges, we propose a framework to predict the DSD of a liquid spray based on data as a function of the spray angle, the Reynolds number, and the Weber number. Second, to guide the design of liquid atomizers, the model accurately predicts the volume of fluid contained in drops of specific sizes while providing uncertainty estimation. To do so, we propose a Gaussian process regression (GPR) model, which infers the DSD and its uncertainty form the knowledge of its integrals and of its first moment, i.e., the mean drop diameter. Third, we deploy multiple GPR models to estimate these quantities at arbitrary points of the design space from data obtained from a large number of numerical simulations of a flat fan spray. The kernel used for reconstructing the DSD incorporates prior physical knowledge, which enables the prediction of sharply peaked and heavy-tailed distributions. Fourth, we compare our method with a benchmark approach, which estimates the DSD by interpolating the frequency polygon of the binned drops with a GPR. We show that our integral approach is significantly more accurate, especially in the tail of the distribution (i.e., large, rare drops), and it reduces the bias of the density estimator by up to 10 times. Finally, we discuss physical aspects of the model's predictions and interpret them against experimental results from the literature. This work opens opportunities for modeling drop size distribution in multiphase flows from data
Multiple-scale thermo-acoustic stability analysis of a coaxial jet combustor
In this paper, asymptotic multiple-scale methods are used to formulate a mathematically consistent set of thermo-acoustic equations in the low-Mach number limit for linear stability analysis. The resulting sets of nonlinear equations for hydrodynamics and acoustics are two-way coupled. The coupling strength depends on which multiple scales are used. The double-time-double-space (2T-2S), double-time-single-space (2T-1S) and single-time-double-space (1T-2S) limits are revisited, derived and linearized. It is shown that only the 1T-2S limit produces a two-way coupled linearized system. Therefore this limit is adopted and implemented in a finite-element solver. The methodology is applied to a coaxial jet combustor. By using an adjoint method and introducing the intrinsic sensitivity, (i) the interaction between the acoustic and hydrodynamic subsystems is calculated and (ii) the role of the global acceleration term, which is the coupling term from the acoustics to the hydrodynamics, is analyzed. For the confined coaxial jet diffusion flame studied here, (i) the growth rate of the thermo-acoustic oscillations is found to be more sensitive to small changes in the hydrodynamic field around the flame and (ii) increasing the global acceleration term is found to be stabilizing in agreement with the Rayleigh Criterion.This is the accepted manuscript. The final version is available at http://www.sciencedirect.com/science/article/pii/S1540748916300670
On the integrability of stationary and restricted flows of the KdV hierarchy.
A bi--Hamiltonian formulation for stationary flows of the KdV hierarchy is
derived in an extended phase space. A map between stationary flows and
restricted flows is constructed: in a case it connects an integrable
Henon--Heiles system and the Garnier system. Moreover a new integrability
scheme for Hamiltonian systems is proposed, holding in the standard phase
space.Comment: 25 pages, AMS-LATEX 2.09, no figures, to be published in J. Phys. A:
Math. Gen.
Reduction of bihamiltonian systems and separation of variables: an example from the Boussinesq hierarchy
We discuss the Boussinesq system with stationary, within a general
framework for the analysis of stationary flows of n-Gel'fand-Dickey
hierarchies. We show how a careful use of its bihamiltonian structure can be
used to provide a set of separation coordinates for the corresponding
Hamilton--Jacobi equations.Comment: 20 pages, LaTeX2e, report to NEEDS in Leeds (1998), to be published
in Theor. Math. Phy
Applications of Temperley-Lieb algebras to Lorentz lattice gases
Motived by the study of motion in a random environment we introduce and
investigate a variant of the Temperley-Lieb algebra. This algebra is very rich,
providing us three classes of solutions of the Yang-Baxter equation. This
allows us to establish a theoretical framework to study the diffusive behaviour
of a Lorentz Lattice gas. Exact results for the geometrical scaling behaviour
of closed paths are also presented.Comment: 10 pages, latex file, one figure(by request
Engineering Silicon Nanocrystals: Theoretical study of the effect of Codoping with Boron and Phosphorus
We show that the optical and electronic properties of nanocrystalline silicon
can be efficiently tuned using impurity doping. In particular, we give
evidence, by means of ab-initio calculations, that by properly controlling the
doping with either one or two atomic species, a significant modification of
both the absorption and the emission of light can be achieved. We have
considered impurities, either boron or phosphorous (doping) or both (codoping),
located at different substitutional sites of silicon nanocrystals with size
ranging from 1.1 nm to 1.8 nm in diameter. We have found that the codoped
nanocrystals have the lowest impurity formation energies when the two
impurities occupy nearest neighbor sites near the surface. In addition, such
systems present band-edge states localized on the impurities giving rise to a
red-shift of the absorption thresholds with respect to that of undoped
nanocrystals. Our detailed theoretical analysis shows that the creation of an
electron-hole pair due to light absorption determines a geometry distortion
that in turn results in a Stokes shift between adsorption and emission spectra.
In order to give a deeper insight in this effect, in one case we have
calculated the absorption and emission spectra going beyond the single-particle
approach showing the important role played by many-body effects. The entire set
of results we have collected in this work give a strong indication that with
the doping it is possible to tune the optical properties of silicon
nanocrystals.Comment: 14 pages 19 figure
Hamiltonian structure of real Monge-Amp\`ere equations
The real homogeneous Monge-Amp\`{e}re equation in one space and one time
dimensions admits infinitely many Hamiltonian operators and is completely
integrable by Magri's theorem. This remarkable property holds in arbitrary
number of dimensions as well, so that among all integrable nonlinear evolution
equations the real homogeneous Monge-Amp\`{e}re equation is distinguished as
one that retains its character as an integrable system in multi-dimensions.
This property can be traced back to the appearance of arbitrary functions in
the Lagrangian formulation of the real homogeneous Monge-Amp\`ere equation
which is degenerate and requires use of Dirac's theory of constraints for its
Hamiltonian formulation. As in the case of most completely integrable systems
the constraints are second class and Dirac brackets directly yield the
Hamiltonian operators. The simplest Hamiltonian operator results in the
Kac-Moody algebra of vector fields and functions on the unit circle.Comment: published in J. Phys. A 29 (1996) 325
Coisotropic deformations of algebraic varieties and integrable systems
Coisotropic deformations of algebraic varieties are defined as those for
which an ideal of the deformed variety is a Poisson ideal. It is shown that
coisotropic deformations of sets of intersection points of plane quadrics,
cubics and space algebraic curves are governed, in particular, by the dKP,
WDVV, dVN, d2DTL equations and other integrable hydrodynamical type systems.
Particular attention is paid to the study of two- and three-dimensional
deformations of elliptic curves. Problem of an appropriate choice of Poisson
structure is discussed.Comment: 17 pages, no figure
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