scholarly journals Material equations in electrodynamics of medium consisting of two-level emitters

2019 ◽  
Vol 27 (1) ◽  
pp. 9-18 ◽  
Author(s):  
S. F. Lyagushyn ◽  
A. I. Sokolovsky ◽  
S. A. Sokolovsky
Keyword(s):  
Correlation Functions ◽  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Dipole Moments ◽  
Closed Set ◽  
Material Parameters ◽  
Material Equations ◽  
Binary Correlation ◽  
Spatial System ◽  
Interaction Term

The process of self-ordering in the famous Dicke model was studied in the framework of eliminating the boson variables. But the reduced description method enables us to obtain also the picture of electromagnetic field evolution provided field amplitudes and correlation functions are included into the number of reduced description parameters. In the Dicke Hamiltonian structure the interaction term includes the operators of emitter dipole moments or dipole moment density (polarization) since a spatial system is under consideration. Thus operator evolution equations are based on using such operators and their derivatives. The chain of evolution equations for averaged field amplitudes and binary correlation functions are obtained with using the statistical operator calculated in a perturbation theory in quasispin-photon interaction assumed to be small. The problem of chain decoupling does not arise since at any step we have a closed set of equations. The sets should be solved on the basis of material equations for current density and their generalizations for more complicated correlation functions. The way to constructing such equations and estimating the material parameters which are necessary for the numerical modeling of the development of correlations is discussed in the paper.

(1+1)-dimensional m-cKdV, g-cKdV integrable systems, and (2+1)-dimensional m-cKdV hierarchy

2008 ◽  
Vol 86 (12) ◽  
pp. 1367-1380 ◽  
Author(s):  
Y Zhang ◽  
H Tam
Keyword(s):  
Integrable Systems ◽  
Linear Functional ◽  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Integrable Model ◽  
Lax Pair ◽  
The Self ◽  
Hamiltonian Structures ◽  
Yang Mills ◽  
Higher Dimensional

A few isospectral problems are introduced by referring to that of the cKdV equation hierarchy, for which two types of integrable systems called the (1 + 1)-dimensional m-cKdV hierarchy and the g-cKdV hierarchy are generated, respectively, whose Hamiltonian structures are also discussed by employing a linear functional and the quadratic-form identity. The corresponding expanding integrable models of the (1 + 1)-dimensional m-cKdV hierarchy and g-cKdV hierarchy are obtained. The Hamiltonian structure of the latter one is given by the variational identity, proposed by Ma Wen-Xiu as well. Finally, we use a Lax pair from the self-dual Yang–Mills equations to deduce a higher dimensional m-cKdV hierarchy of evolution equations and its Hamiltonian structure. Furthermore, its expanding integrable model is produced by the use of a enlarged Lie algebra.PACS Nos.: 02.30, 03.40.K

SCALE DEPENDENCE OF TWIST-3 CORRELATION FUNCTIONS

2011 ◽  
Vol 04 ◽  
pp. 146-156
Author(s):  
Zhong-Bo Kang ◽  
Jian-Wei Qiu
Keyword(s):  
Correlation Functions ◽  
Evolution Equations ◽  
Scale Dependence ◽  
Leading Order ◽  
Coupling Constant ◽  
Parton Distributions ◽  
Spin Asymmetries ◽  
Single Spin ◽  
Single Spin Asymmetries ◽  
Similarities And Differences

In this talk, we introduce two sets of twist-3 quark-gluon correlation functions that are relevant to transverse single spin asymmetries, and present corresponding evolution equations at the leading order in strong coupling constant, αs. The similarities and differences between the evolution of the leading power parton distributions and that of the twist-3 multiparton correlation functions are also discussed.

Wave modulation: the geometry, kinematics, and dynamics of surface-wave packets

2016 ◽  
Vol 803 ◽  
pp. 292-312 ◽  
Author(s):  
N. E. Pizzo ◽  
W. Kendall Melville
Keyword(s):  
Evolution Equations ◽  
Wave Packets ◽  
A Priori ◽  
Analytic Solutions ◽  
Kinematics And Dynamics ◽  
Linear Dispersion ◽  
Closed Set ◽  
Weakly Nonlinear ◽  
Numerical Predictions ◽  
Variational Structure

We examine the geometry, kinematics, and dynamics of weakly nonlinear narrow-banded deep-water wave packets governed by the modified nonlinear Schrödinger equation (Dysthe, Proc. R. Soc. Lond. A., vol. 369, 1979, pp. 105–114; MNLSE). A new derivation of the spatial MNLSE, by a direct application of Whitham’s method, elucidates its variational structure. Using this formalism, we derive a set of conserved quantities and moment evolution equations. Next, by examining the MNLSE in the limit of vanishing linear dispersion, analytic solutions can be found. These solutions then serve as trial functions, which when substituted into the moment evolution equations form a closed set of equations, allowing for a qualitative and quantitative examination of the MNLSE without resorting to numerically solving the full equation. To examine the theory we consider initially symmetric, chirped and unchirped wave packets, chosen to induce wave focusing and steepening. By employing the ansatz for the trial function discussed above, we predict, a priori, the evolution of the packet. It is found that the speed of wave packets governed by the MNLSE depends on their amplitude, and in particular wave groups speed up as they focus. Next, we characterize the asymmetric growth of the wave envelope, and explain the steepening of the forward face of the initially symmetric wave packet. As the packet focuses, its variance decreases, as does the chirp of the signal. These theoretical results are then compared with the numerical predictions of the MNLSE, and agreement for small values of fetch is found. Finally, we discuss the results in the context of existing theoretical, numerical and laboratory studies.

INTEGRABLE HAMILTONIAN HIERARCHIES ASSOCIATED WITH THE EQUATION OF HEAT CONDUCTION

2010 ◽  
Vol 24 (14) ◽  
pp. 1573-1594 ◽  
Author(s):  
YUFENG ZHANG ◽  
HONWAH TAM ◽  
JIANQIN MEI
Keyword(s):  
Heat Conduction ◽  
Quadratic Form ◽  
Lie Algebra ◽  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Arbitrary Parameter ◽  
Higher Dimensional ◽  
Soliton Hierarchy ◽  
Equation Of Heat Conduction ◽  
Hamiltonian Functions

Using a 4-dimensional Lie algebra g, an isospectral Lax pair is introduced, whose compatibility condition is equivalent to a soliton hierarchy of evolution equations with three components of potential functions. Its Hamiltonian structure is obtained by employing the quadratic-form identity proposed by Guo and Zhang. In order to obtain explicit Hamiltonian functions, a detailed computing formula for the constant appearing in the quadratic-form identity is obtained. One type of reduction equations of the hierarchy is also produced, which is further reduced to the standard equation of heat conduction. By introducing a loop algebra of the Lie algebra g, we obtain a soliton hierarchy with an arbitrary parameter which can be reduced to the previous equation hierarchy obtained, whose quasi-Hamiltonian structure is also worked out by the quadratic-form identity. Finally, we extend the Lie algebra g into a higher-dimensional Lie algebra so that a new integrable Hamiltonian hierarchy, which comprise integrable couplings, is produced; its reduced equations in particular contain two arbitrary parameters.

scholarly journals EVOLUTION EQUATIONS OF TWIST-THREE PARTON DISTRIBUTIONS

2011 ◽  
Vol 04 ◽  
pp. 157-167
Author(s):  
BJÖRN PIRNAY
Keyword(s):  
Recent Work ◽  
Correlation Functions ◽  
Evolution Equations ◽  
Spin Asymmetry ◽  
Scale Dependence ◽  
Transverse Spin ◽  
Collinear Factorization ◽  
Parton Distributions ◽  
Gluon Momentum

We report on a recent work concerning the scale dependence of twist-three correlation functions relevant for the single transverse spin asymmetry in the framework of collinear factorization. Evolution equations are presented for both the flavor–nonsinglet and flavor–singlet distributions. Our results do not agree with previous calculations of the evolution in the limit of vanishing gluon momentum. Possible sources for this discrepancy are identified.

On the Hamiltonian structure of evolution equations

1980 ◽  
Vol 88 (1) ◽  
pp. 71-88 ◽  
Author(s):  
Peter J. Olver
Keyword(s):  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Formal Theory ◽  
Natural Generalization ◽  
Line Integral ◽  
Theory Of Evolution ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Integral Invariants ◽  
Generalized Symmetries

AbstractThe theory of evolution equations in Hamiltonian form is developed by use of some differential complexes arising naturally in the formal theory of partial differential equations. The theory of integral invariants is extended to these infinite-dimensional systems, providing a natural generalization of the notion of a conservation law. A generalization of Noether's theorem is proved, giving a one-to-one correspondence between one-parameter (generalized) symmetries of a Hamiltonian system and absolute line integral invariants. Applications include a new solution to the inverse problem of the calculus of variations, an elementary proof and generalization of a theorem of Gel'fand and Dikiî on the equality of Lie and Poisson brackets for Hamiltonian systems, and a new hierarchy of conserved quantities for the Korteweg–de Vries equation.

BIRTH OF ANOMALOUS SCALING IN A MODEL OF HYDRODYNAMIC TURBULENCE WITH A TUNABLE PARAMETER

Fractals ◽  
2002 ◽  
Vol 10 (03) ◽  
pp. 291-296
Author(s):  
D. PIEROTTI ◽  
V. S. L'VOV ◽  
A. POMYALOV ◽  
I. PROCACCIA
Keyword(s):  
Numerical Simulations ◽  
Correlation Functions ◽  
Order Structure ◽  
Anomalous Scaling ◽  
Scale Invariant ◽  
Scaling Exponents ◽  
Closed Set ◽  
Hydrodynamic Turbulence ◽  
Turbulent Field ◽  
Random Components

We introduce a model of hydrodynamic turbulence with a tunable parameter ε, which represents the ratio between deterministic and random components in the coupling between N identical copies of the turbulent field. To compute the anomalous scaling exponents ζn (of the nth order structure functions) for chosen values of ε, we consider a systematic closure procedure for the hierarchy of equations for the n-order correlation functions, in the limit N → ∞. The parameter ε regularizes the closure procedure, in the sense that discarded terms are of higher order in ε compared to those retained. It turns out that after the terms of O(1), the first nonzero terms are O(ε4). Within this ε-controlled procedure, we have a finite and closed set of scale-invariant equations for the 2nd and 3rd order statistical objects of the theory. This set of equations retains all terms of O(1) and O(ε4) and neglects terms of O(ε6). On this basis, we expect anomalous corrections δ ζn in the scaling exponents ζn to increase with εn. This expectation is confirmed by extensive numerical simulations using up to 25 copies and 28 shells for various values of εn. The simulations demonstrate that in the limit N → ∞, the scaling is normal for ε < ε cr with ε cr ≈ 0.6. We observed the birth of anomalous scaling at ε = ε cr with [Formula: see text] according to our expectation.

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