Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

The concept of parity-time symmetric systems is rooted in non-Hermitian quantum mechanics where complex potentials obeying this symmetry could exhibit real spectra. The concept has applications in many fields of physics, e.g., in optics, metamaterials, acoustics, Bose-Einstein condensation, electronic circuitry, etc. The inclusion of nonlinearity has led to a number of new phenomena for which no counterparts exist in traditional dissipative systems. Several examples of nonlinear parity-time symmetric systems in different physical disciplines are presented and their implications discussed.

Nonlinear waves in PT -symmetric systems

One of the challenges of the modern photonics is to develop all-optical devices enabling increased speed and energy efficiency for transmitting and processing information on an optical chip. It is believed that the recently suggested Parity-Time (PT) symmetric photonic systems with alternating regions of gain and loss can bring novel functionalities. In such systems, losses are as important as gain and, depending on the structural parameters, gain compensates losses. Generally, PT systems demonstrate nontrivial non-conservative wave interactions and phase transitions, which can be employed for signal filtering and switching, opening new prospects for active control of light. In this review, we discuss a broad range of problems involving nonlinear PT-symmetric photonic systems with an intensity-dependent refractive index. Nonlinearity in such PT symmetric systems provides a basis for many effects such as the formation of localized modes, nonlinearly-induced PT-symmetry breaking, and all-optical switching. Nonlinear PT-symmetric systems can serve as powerful building blocks for the development of novel photonic devices targeting an active light control.

Nonlinear switching and solitons in PT-symmetric photonic systems

1. Introduction.- 1.1 The Fermi-Pasta-Ulam Problem.- 1.2 Henon-Heiles Calculation.- 1.3 Discovery of Solitons.- 1.4 Dual Systems.- 2. The Lattice with Exponential Interaction.- 2.1 Finding of an Integrable Lattice.- 2.2 The Lattice with Exponential Interaction.- 2.3 Periodic Solutions.- 2.4 Solitary Waves.- 2.5 Two-Soliton Solutions.- 2.6 Hard-Sphere Limit.- 2.7 Continuum Approximation and Recurrence Time.- 2.8 Applications and Extensions.- 2.9 Poincare Mapping.- 2.10 Conserved Quantities.- 3. The Spectrum and Construction of Solutions.- 3.1 Matrix Formalism.- 3.2 Infinite Lattice.- 3.3 Scattering and Bound States.- 3.4 The Gel'fand-Levitan Equation.- 3.5 The Initial Value Problem.- 3.6 Soliton Solutions.- 3.7 The Relationship Between the Conserved Quantities and the Transmission Coefficient.- 3.8 Extensions of the Equations of Motion and the Kac-Moerbeke System.- 3.9 The Backlund Transformation.- 3.10 A Finite Lattice.- 3.11 Continuum Approximation.- 4. Periodic Systems.- 4.1 Discrete Hill's Equation.- 4.2 Auxiliary Spectrum.- 4.3 Relation Between ?j (k) and ?j (0).- 4.4 Related Integrals on the Riemann Surface.- 4.5 Solution to the Inverse Problem.- 4.6 Time Evolution.- 4.7 A Simple Example (A Cnoidal Wave).- 4.8 Periodic System of Three-Particles.- 5. Application of the Hamilton-Jacobi Theory.- 5.1 Canonically Conjugate Variables.- 5.2 Action Variables.- 6. Recent Advances in the Theory of Nonlinear Lattices.- 6.1 The KdV Equation as a Limit of the TL Equation.- 6.2 Interacting Soliton Equations.- 6.3 Integrability.- 6.4 Generalization of the TL Equation.- 6.5 Two-Dimensional TL.- 6.6 Bethe Ansatz.- 6.7 The Thermodynamic Limit.- 6.8 Hierarchy of Nonlinear Equations.- 6.9 Some Numerical Results.- Appendices.- Simplified Answers to Main Problems.- References.- List of Authors Cited in Text.

Theory of nonlinear lattices

The purpose of this contribution is to develop a controllability method for linear discrete systems with constant coefficients and with pure delay. To do this, a representation of solutions with the aid of a discrete matrix delayed exponential is used. Such an approach leads to new conditions of controllability. Except for a criterion of relative controllability, a control function is constructed as well.

Controllability of Linear Discrete Systems with Constant Coefficients and Pure Delay

Here, we give explicit formulae for solutions of some systems of difference equations, which extend some very particular recent results in the literature and give natural explanations for them, which were omitted in the previous literature.

/pdf/on-some-solvable-difference-equations-and-systems-of-2vx54vxh7h.pdf

On Some Solvable Difference Equations and Systems of Difference Equations

Representation of solutions of discrete delayed system x(k+1)=Ax(k)+Bx(k−m)+f(k) with commutative matrices

We show that the system of three difference equations , , and , , where all elements of the sequences , , , , , and initial values , , , , are real numbers, can be solved. Explicit formulae for solutions of the system are derived, and some consequences on asymptotic behavior of solutions for the case when coefficients are periodic with period three are deduced.

/pdf/on-a-third-order-system-of-difference-equations-with-3fxms89bgn.pdf

Josef Diblík

Papers

Controllability of Linear Discrete Systems with Constant Coefficients and Pure Delay

On Some Solvable Difference Equations and Systems of Difference Equations

Representation of solutions of discrete delayed system x(k+1)=Ax(k)+Bx(k−m)+f(k) with commutative matrices

On a Third-Order System of Difference Equations with Variable Coefficients

Exponential stability of linear discrete systems with constant coefficients and single delay