Fano resonance and wave transmission through a chain structure with an isolated ring composed of defects

We consider a discrete model that describes a linear chain of particles coupled to an isolated ring composed of N defects.This simple system can be regarded as a generalization of the familiar Fano-Anderson model.It can be used to model discrete networks of coupled defect modes in photonic crystals and simple waveguide arrays in two-dimensional lattices.The analytical result of the transmission coefficient is obtained,along with the conditions for perfect reflections and transmissions due to either destructive or constructive interferences.Using a simple example,we further investigate the relationship between the resonant frequencies and the number of defects N,and study how to affect the numbers of perfect reflections and transmissions.In addition,we demonstrate how these resonance transmissions and refections can be tuned by one nonlinear defect of the network that possesses a nonlinear Kerr-like response.     

Stability and instability of nonlinear defect states in the coupled mode equations—Analytical and numerical study

Coupled backward and forward wave amplitudes of an electromagnetic field propagating in a periodic and nonlinear medium at Bragg resonance are governed by the nonlinear coupled mode equations (NLCME). This system of PDEs, similar in structure to the Dirac equations, has gap soliton solutions that travel at any speed between 0 and the speed of light. A recently considered strategy for spatial trapping or capture of gap optical soliton light pulses is based on the appropriate design of localized defects in the periodic structure. Localized defects in the periodic structure give rise to defect modes, which persist as nonlinear defect modes as the amplitude is increased. Soliton trapping is the transfer of incoming soliton energy to nonlinear defect modes. To serve as targets for such energy transfer, nonlinear defect modes must be stable. We therefore investigate the stability of nonlinear defect modes. Resonance among discrete localized modes and radiation modes plays a role in the mechanism for stability and instability, in a manner analogous to the nonlinear Schrödinger/Gross-Pitaevskii (NLS/GP) equation. However, the nature of instabilities and how energy is exchanged among modes is considerably more complicated than for NLS/GP due, in part, to a continuous spectrum of radiation modes which is unbounded above and below. In this paper we (a) establish the instability of branches of nonlinear defect states which, for vanishing amplitude, have a linearization with eigenvalues embedded within the continuous spectrum, (b) numerically compute, using Evans function, the linearized spectrum of nonlinear defect states of an interesting multiparameter family of defects, and (c) perform direct time-dependent numerical simulations in which we observe the exchange of energy among discrete and continuum modes.     

Transport in a disordered one-dimensional system: A fractal view

We reexamine the calculation of the transmission coefficient of a random array ofN isotopic defects in an otherwise perfect, harmonic, one-dimensional crystal lattice. The thermal conductivity of this model system has been studied under steady state conditions in which there is a kinetic temperature difference across, and an associated energy flux through, the array of defects. An exact expression for the transmission coefficient is obtained in terms of the magnitude of anNth-order determinant. Rubin reduced the evaluation of the determinant to the evaluation of a sequence ofN–1 nonlinear transformations drawn from a set of transformations parametrized by the nearest-neighbor spacing of the isotopic defects. These transformations are self-inverse and provide an example of what Mandelbrot has termed aself-inverse fractal. The variety of limiting distributions of values obtained under these transformations will be illustrated.     

A perfect lattice approach to nonstoichiometry

Although nonstoichiometry implies the presence of defects in a structure and hence a departure from perfect crystallinity, it is sometimes found that the defects are ordered or, at least, show a tendency to order. In these cases, it becomes possible to model the nonstoichiometry by considering a supercell and using perfect lattice techniques. We illustrate the viability of this approach with two examples. Firstly, we show how factors controlling the long-range ordering of extended defects in transition-metal oxides may be elucidated; indeed, we find that an adequate treatment of this phenomenon requires calculations on supercells. Secondly, we discuss how perfect lattice calculations may be used, in conjunction with diffraction data, to examine possible vacancy ordering schemes in the oxidation of magnetite, Fe₃O₄, to maghemite, the defect spinel structured γ-Fe₂O₃.     

Linear bosonic quantum field theories arising from causal variational principles

We describe discrete symmetries of two-dimensional Yang–Mills theory with gauge group G associated with outer automorphisms of G, and their corresponding defects. We show that the gauge theory partition function with defects can be computed as a path integral over the space of twisted G-bundles and calculate it exactly. We argue that its weak-coupling limit computes the symplectic volume of the moduli space of flat twisted G-bundles on a surface. Using the defect network approach to generalised orbifolds, we gauge the discrete symmetry and construct the corresponding orbifold theory, which is again two-dimensional Yang–Mills theory but with gauge group given by an extension of G by outer automorphisms. With the help of the orbifold completion of the topological defect bicategory of two-dimensional Yang–Mills theory, we describe the reverse orbifold using a Wilson line defect for the discrete gauge symmetry. We present our results using two complementary approaches: in the lattice regularisation of the path integral, and in the functorial approach to area-dependent quantum field theories with defects via regularised Frobenius algebras.

     

Multi-place physics and multi-place nonlocal systems

Multi-place nonlocal systems have attracted attention from many scientists. In this paper, we mainly review the recent progresses on two-place nonlocal systems (Alice-Bob systems) and four-place nonlocal models. Multi-place systems can firstly be derived from many physical problems by using a multiple scaling method with a discrete symmetry group including parity, time reversal, charge conjugates, rotations, field reversal and exchange transformations. Multi-place nonlocal systems can also be derived from the symmetry reductions of coupled nonlinear systems via discrete symmetry reductions. On the other hand, to solve multi-place nonlocal systems, one can use the symmetry-antisymmetry separation approach related to a suitable discrete symmetry group, such that the separated systems are coupled local ones. By using the separation method, all the known powerful methods used in local systems can be applied to nonlocal cases. In this review article, we take two-place and four-place nonlocal nonlinear Schrödinger (NLS) systems and Kadomtsev-Petviashvili (KP) equations as simple examples to explain how to derive and solve them. Some types of novel physical and mathematical points related to the nonlocal systems are especially emphasized.     

A class of solvable coupled nonlinear oscillators with amplitude independent frequencies

Existence of amplitude independent frequencies of oscillation is an unusual property for a nonlinear oscillator. We find that a class of N coupled nonlinear Liénard type oscillators exhibit this interesting property. We show that a specific subset can be explicitly solved from which we demonstrate the existence of periodic and quasiperiodic solutions. Another set of N coupled nonlinear oscillators, possessing the amplitude independent nature of frequencies, is almost integrable in the sense that the system can be reduced to a single nonautonomous first order scalar differential equation which can be easily integrated numerically.     

Solutions of several coupled discrete models in terms of Lamé polynomials of order one and two

Coupled discrete models abound in several areas of physics. Here we provide an extensive set of exact quasiperiodic solutions of a number of coupled discrete models in terms of Lamé polynomials of order one and two. Some of the models discussed are: (i) coupled Salerno model, (ii) coupled Ablowitz–Ladik model, (iii) coupled saturated discrete nonlinear Schrödinger equation, (iv) coupled ? ⁴ model and (v) coupled ? ⁶ model. Furthermore, we show that most of these coupled models in fact also possess an even broader class of exact solutions.     

Properties of supersymmetric integrable systems of KP type

The recently proposed supersymmetric extensions of reduced Kadomtsev-Petviashvili (KP) integrable hierarchies in N = 1, 2 superspace are shown to contain in the purely bosonic limit new types of ordinary non-supersymmetric integrable systems. The latter are coupled systems of several multi-component non-linear Schr?dinger-like hierarchies whose basic nonlinear evolution equations contain additional quintic and higher-derivative nonlinear terms. Also, we obtain the N = 2 supersymmetric extension of Toda chain model as Darboux-B?cklund orbit of the simplest reduced N = 2 super-KP hierarchy and find its explicit solution. Received 13 September 2001 Published online 2 October 2002 RID="a" ID="a"e-mail: nissimov@inrne.bas.bg RID="b" ID="b"e-mail: svetlana@inrne.bas.bg     

金红石相和锐钛矿相TiO2本征缺陷的第一性原理计算

采用第一性原理的计算方法, 分别研究了金红石相和锐钛矿相TiO₂各种缺陷态形成的类型, 以及几何结构、生长气氛和Fermi能级位置对缺陷形成能的影响, 从理论上预测产生点缺陷的实验条件. 重点是讨论带电点缺陷的形成能, 并对结果进行适当修正. 研究发现, 本征缺陷的类型和浓度对 TiO₂的性能有一定的影响: 在富O条件下, TiO₂容易形成V_Ti(Ti空位)缺陷; 在富Ti条件下, TiO₂的Ti_i⁴⁺和V_O(O空位)缺陷将大量出现, 形成Schottky缺陷.