The discrete Korteweg-de Vries equation

We review the different aspects of integrable discretizations in space and time of the Korteweg-de Vries equation, including Miura transformations to related integrable difference equations, connections to integrable mappings, similarity reductions and discrete versions of Painlevé equations as well as connections to Volterra systems.     

On the discrete spectrum of a perturbed Wiener-Hopf integral operator

Translated from Matematicheskie Zametki, Vol. 51, No. 1, pp. 102–113, January, 1992.     

Wiener-Hopf equation on the half-axis in the Nevanlinna and Smirnov algebras

Wiener-Hopf equations are investigated on the half- axis in the Nevanlinna and Smirnov classes. The solvability of these equations is proved under natural restrictions and formulas describing the solutions are found.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 67–81, 1989.     

Wiener-Hopf operators on a subsemigroup of a discrete torsion free abelian group

In the paper Wiener-Hopf operators on a semigroup of nonnegative elements of a linearly quasi-ordered torsion free Abelian group are considered. Wiener-Hopf factorization of an invertible element of the group algebra is constructed, notions of a topological index and a factor index are introduced. It turns out that the set of factor indices for invertible elements of the group algebra is a linearly ordered group. It is shown that Wiener-Hopf operator with an invertible symbol is an one-side invertible operator and its invertibility properties are defined by the sign of the factor index of its symbol. Groups on which there exist nontrivial Fredholm Wiener-Hopf operators are described. As an example, all linear quasi-orders on the group ⁿ are found and corresponding Wiener-Hopf operators are considered.     

Wiener-Hopf equation: Kernels representable as a superposition of complex-valued exponents

The paper studies the Wiener-Hopf equations with kernels representable as superposition of complex-valued exponents. Such kernels arise in the kinetic gas theory, in the radiation transfer, etc. By application of a special, three-factor expansion of the initial uninvertible operator, the solution of the considered equation is reduced to those of two simple Volterra equations and a Wiener-Hopf integral equation with a contractive operator. A structural existence theorem is proved.     

An asymptotic property of the solution to the homogeneous generalized Wiener-Hopf equation

We consider the homogeneous generalized Wiener-Hopf equation

Spectral factorization of rectangular rational matrix functions with application to discrete Wiener-Hopf equations

The properties of a discrete Wiener-Hopf equation are closely related to the factorization of the symbol of the equation. We give a necessary and sufficient condition for existence of a canonical Wiener-Hopf factorization of a possibly nonregular rational matrix function W relative to a contour which is a positively oriented boundary of a region in the finite complex plane. The condition involves decomposition of the state space in a minimal realization of W and, if it is satisfied, we give explicit formulas for the factors. The results are generalized by means of centered realizations to arbitrary rational matrix functions. The proposed approach can be used to solve discrete Wiener-Hopf equations whose symbols are rational matrix functions which admit canonical factorization relative to the unit circle.     

Tsirel'son's equation in discrete time

Summary Motivated by Tsirel'son's equation in continuous time, a similar stochastic equation indexed by discrete negative time is discussed in full generality, in terms of the law of a discrete time noise. When uniqueness in law holds, the unique solution (in law) is not strong; moreover, when there exists a strong solution, there are several strong solution. In general, for any time,n, the -field generated by the past of a solution up to timen is shown to be equal, up to negligible sets, to the -field generated by the 3 following components: the infinitely remote past of the solution, the past to the noise up to timen, together with an adequate independent complement.     

Some solutions of discrete sine-Gordon equation

In this paper, a series of exact solutions of discrete sine-Gordon equation are obtained by the different transformations and symbolic computation.     

The application of Padeapproximants to Wiener-Hopf factorization

The key step in the solution of a Wiener–Hopf equation is the decomposition of the Fourier transform of the kernel,which is a function of a complex variable, say, into a productof two terms. One is singularity and zero free in an upperregion of the -plane, and the other singularity and zero freein an overlapping lower region. Each product factor can beexpressed in terms of a Cauchy-type integral formula, but thisform presents difficulties due to the speed of its evaluationand numerical problems caused by singularities near the integrationcontour. Other representations are available in special cases,for instance an infinite product form for meromorphic functions,but not in general. To overcome these problems, several approximatemethods for decomposing the transformed kernels have been suggested.However, whilst these offer simple explicit expressions, theirforms tend to have been derived in an ad hoc fashion and todate have only mediocre accuracy (of order one per cent orso). A new method for approximating Wiener–Hopf kernelsis offered in this article which employs Padéapproximants.These have the advantage of offering very simple approximatefactors of Fourier transformed kernels which are found to beextremely accurate for modest computational effort. Further,the derivation of the factors is algorithmic and thereforerequires little effort, and the Padénumber is a convenientparameter with which to reduce errors to within set targetvalues. The paper demonstrates the efficacy of the approachon several model kernels, and numerical results presented hereinconfirm theoretical predictions regarding convergence to theexact results, etc. The relationship between the present methodand earlier approximate schemes is discussed. Received 7 February, 1998. Revised 18 February, 1999.     

The vector-valued Wiener-Hopf equation for a mixed boundary-value problem of diffraction at a wedge intersected by a half-plane

For a mixed boundary-value problem in a nonregular region we obtain the vector-valued Wiener-Hopf equation, which is then reduced to infinite systems of linear algebraic equations using the factorization method and Liouville's theorem. It then becomes possible to solve the equation with prescribed precision for arbitrary values of the parameters of the problem. In the stationary case the solution is obtained in closed form.Translated fromMatematichni Metodi ta Fiziko-Mechanichni Polya, Vol. 40, No. 3, 1997, pp. 87–92.     

The convergence of fully discrete Galerkin approximations of the Rosenau equation

In this paper, we shall analyze the fully discrete Galerkin type approximations to solutions of the Rosenau equation. We provide the numerical results of several cases.     

Entire solutions for a discrete diffusive equation

We study entire solutions of a discrete diffusive equation with bistable nonlinearity. It is well known that there are three different wavefronts connecting any two of those three equilibria, say, 0,a,1. We construct three different types of entire solutions. The first one is a solution which behaves as two opposite wavefronts (connecting 0 and 1) of the same positive speed approaching each other from both sides of the real line. The second one is a solution which behaves as two different wavefronts (connecting a and one of {0,1}) approaching each other from both sides of the real line and converging to the wavefront connecting 0 and 1. The third one is a solution which behaves as a wavefront connecting a and 0 and a wavefront connecting 0 and 1 approaching each other from both sides of the real line.     

A discrete version of the Landau-Lifshits equation

A discrete version of the Landau-Lifshits equation from the theory of ferromagnetism is investigated within the framework of the method of the inverse-scattering problem. Variations of action-angle type are constructed, and the energy spectrum of the model is described. The procedure of dressing is used to obtain the simplest soliton solution.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 145, pp. 62–71, 1985.