Differential-difference Systems Research Articles

Conservation laws for two (2+1)-dimensional differential–difference systems

Two integrable differential–difference equations are considered. One is derived from the discrete BKP equation and the other is a symmetric (2 + 1)-dimensional Lotka–Volterra equation. An infinite number of conservation laws for the two differential–difference equations are deduced.

Controllability and Identifiability of Multi-Delay Dynamic Systems

Complete controllability and identifiability of autonomous linear differential-difference systems with comparable delays are studied. A continuous current state estimator is constructed from output measurements. A method for computing the stabilizing control is described. Results are extended to parabolic-type autonomous linear partial differential dynamic delay systems.

Soliton Solutions of Two (2+1)-dimensional Differential-Difference Systems

In this paper, multi-soliton solutions of two (2+1)-dimensional systems [D l sinh(1 2D n ) + D 3 x sinh(1 2D n )]f n . f n = 0 and [D x D t cosh(1 2 D n ) + D t sinh(1 2 D n )]f n . f n = 0 are derived and expressed by pfaffians.

Asymptotic Behavior of Solutions of Dynamic Equations

We consider linear dynamic systems on time scales, which contain as special cases linear differential systems, difference systems, or other dynamic systems. We give an asymptotic representation for a fundamental solution matrix that reduces the study of systems in the sense of asymptotic behavior to the study of scalar dynamic equations. In order to understand the asymptotic behavior of solutions of scalar linear dynamic equations on time scales, we also investigate the behavior of solutions of the simplest types of such scalar equations, which are natural generalizations of the usual exponential function.

Invariant Submanifolds of the Darboux–Kadomtsev–Petviashvili Chain and an Extension of the Discrete Kadomtsev–Petviashvili Hierarchy

We investigate invariant submanifolds of the so-called Darboux–Kadomtsev–Petviashvili chain. We show that restricting the dynamics to a class of invariant submanifolds yields an extension of the discrete Kadomtsev–Petviashvili hierarchy and intersections of invariant submanifolds yield the Lax description of a wide class of differential–difference systems. We consider self-similar reductions. We show that self-similar substitutions result in purely discrete equations that depend on a finite set of parameters and in equations determining deformations w.r.t. these parameters. We present examples. In particular, we show that the well-known first discrete Painleve equation corresponds to the Volterra chain hierarchy. We derive the equations naturally generalizing the first discrete Painleve equation in the sense that all of them become the first Painleve equation in the continuum limit.

Variable Separation Approach for a Differential-difference Asymmetric Nizhnik-Novikov-Veselov Equation

The multi-linear variable separation approach is applied to a differential-difference asymmetric Nizhnik-Novikov-Veselov equation. It is found that the solution formula ANNV equation is rightly the semi-discrete form of the continuous one which describes some types of special solutions for many (2+1)-dimensional continuous systems. Moreover, it is different but similar to that of a special differential-difference Toda system. Thus abundant semi-discrete localized coherent structures of the ANNV equation are easily constructed by appropriately selecting the arbitrary functions appearing in the final solution formula. A concrete method to construct multiple localized discrete excitations with and without completely elastic interaction properties are discussed. It is found that for some types of bounded solitoff modes, they can exchange solitoffs, move in new different directions, extend the lengths of solitoffs, etc. after finishing their interactions. - PACS numbers: 0230, 0220, 0540

Variable separation approach for a differential-difference system: special Toda equation

A bilinear variable separation approach is used to construct some special solutions for a differential-difference Toda equation. The semi-discrete form of the continuous formula which describes some types of special solutions for many (2 + 1)-dimensional continuous systems is found for a suitable quantity of the differential-difference Toda equation. Thus abundant semi-discrete localized coherent structures are constructed by appropriately selecting the arbitrary functions.

Linear fractal games of pursuit

A general method for solving game problems of pursuit is proposed for dynamical systems with Volterra evolution. The method makes use of resolving functions [1] and the tools of the theory of set-valued mappings. The scheme proposed covers a wide range of functional-differential systems, such as integral, integrodifferential and differential-difference systems of equations defining the dynamics of conflict-controlled processes. A more detailed study is made of game problems for systems with Riemann-Liouville fractional derivatives and regularized Dzhrbashyan-Nersesyan derivatives (“fractal” games). Asymptotic representations of generalized Mittag-Löffler functions are used in the context of the method to establish sufficient conditions for the solvability of game problems.

Explicit solutions of some (2+1)-dimensional differential-difference equations

Some (2+1)-dimensional differential-difference equations are decomposed into two (1+1)-dimensional differential-difference systems in the same hierarchy. With the help of the decomposition and the Darboux transformation, some explicit solutions of these (2+1)-dimensional differential-difference equations are obtained.

Lax pairs and conservation laws for two differential-difference systems

A coupled extended Lotka–Volterra lattice and a special Toda lattice are derived from the existing bilinear equations. Starting from the corresponding bilinear Bäcklund transformation, Lax pairs for these two differential-difference systems are obtained. Furthermore, an infinite number of conservation laws for the differential-difference equations are deduced from the Lax pairs in a systematic way.

Theoretical and experimental aspects of chaos control by time-delayed feedback.

We review recent developments for the control of chaos by time-delayed feedback methods. While such methods are easily applied even in quite complex experimental context the theoretical analysis yields infinite-dimensional differential-difference systems which are hard to tackle. The essential ideas for a general theoretical approach are sketched and the results are compared to electronic circuits and to high power ferromagnetic resonance experiments. Our results show that the control performance can be understood on the basis of experimentally accessible quantities without resort to any model for the internal dynamics.

An Integrable Coupled Toda Equation And Its Related Equation Via Hirota’s Bilinear Approach

A coupled Toda equation and its related equation are derived from 3-coupled bilinear equations. The corresponding Bäcklund transformation and nonlinear superposition formula are presented for the 3-coupled bilinear equations. As an application of the results, solition solutions are derived. Besides, starting from the bilinear BT, Lax pairs for these two differential-difference systems are obtained.

Conservation Laws of the Two-Dimensional Toda Lattice Hierarchy

A novel method of constructing the conservation laws of (1+2)-dimensional differential-difference systems is proposed. By introducing the generalized Riccati equation related to the pseudo-difference operator, we obtain the infinitely many conserved densities and the associated fluxes of the two-dimensional Toda lattice hierarchy. Moreover, this method presents more forms of the conservation laws than the previous approach [K. Kajiwara and J. Satsuma: J. Math. Phys. 32 (1991) 506].

Phase Space Extension in the Analysis of Differential-Difference Systems with Random Input

Stochastic differential-difference systems with constant delay are investigated by a method based on the extension of the phase space by constructing a chain of Fokker–Planck–Kolmogorov equations for the transition distribution densities of phase vectors of increasing dimension. Examples are given to illustrate the application of the method.

INTEGRABLE CHAINS AND HIERARCHIES OF DIFFERENTIAL EVOLUTION EQUATIONS

A class of integrable differential–difference systems is constructed based on auxiliary linear equations defined on sequences of Zakharov–Shabat formal dressing operators. We show that the auxiliary equations are compatible with the evolution equations for the Kadomtsev–Petviashvili (KP) hierarchy. The investigation results are used to elaborate a modified version of Krichever rational reductions for KP hierarchies.

A differential-difference system related to toroidal Lie algebra

We present a novel differential-difference system in (2+1)-dimensional space-time (one discrete, two continuum), arising from Bogoyavlensky's (2+1)-dimensional Korteweg-de Vries hierarchy. Our method is based on the bilinear identity of the hierarchy, which is related to the vertex-operator representation of the toroidal Lie algebra, tor2.

Effects of small delays on stability of singularly perturbed systems

A small delay in the feedback loop of a singularly perturbed system may destabilize it; however, without the delay, it is stable for all small enough values of a singular perturbation parameter ε. Sufficient and necessary conditions for preserving stability, for all small enough values of delay and ε, are obtained in two cases: in the case of delay proportional to ε and in the case of independent delay and ε. In the second case, the sufficient conditions are given in terms of an LMI. A delay-dependent LMI criterion for the stability of singularly perturbed differential–difference systems is derived.

Families of particular solutions to multidimensional partial differential equations

Dressing methods are known as productive tools for construction of the particular solutions to the big class of nonlinear partial differential equations (PDEs) which are integrable by the inverse scattering technique. Recently the modification of the dressing method based on the system of algebraic equations has been suggested which allows us to find the families of particular solutions to certain types of nonintegrable (in classical sense) nonlinear PDEs. This modification represents PDEs as closure reductions of an appropriate differential-difference system. In this article we study the dressing procedure in more detail. Particularly, we consider different families of particular solutions available through the dressing method based on the algebraic system. We give two examples of the differential-difference systems and related PDEs and point to other possible generalizations of the dressing method.

Application of bilinear method to integrable differential-difference equations

Two coupled bilinear equations are considered, and then two new coupled differential-difference systems are found. Also two special reductions of these two systems are studied. By using Hirota's method, Bäcklund transformation and superposition formulae, soliton solutions to these equations are presented.

New integrable differential-difference systems: Lax pairs, bilinearforms and soliton solutions

Two new integrable differential-difference systems with their Lax pairs areproposed. By the dependent variable transformations, these integrable lattices can be transformed into bilinear equations.With the assistance of Mathematica, three-soliton solutions are explicitly obtained. We have also shown that these lattices can be obtained from a special case of the coupled bilinear equationsunder reduction. Furthermore a bilinear Bäcklundtransformation and the corresponding nonlinearsuperposition formula concerning the coupled bilinear equations are presented. Besides, it is also illustrated that the y-flow of these coupled bilinear equationscan be transformed into a latticepreviously derived by the authors.Starting from the corresponding bilinear Bäcklundtransformation, its corresponding Lax pair is obtained.