Related papers: Discrete equations and the singular manifold metho…
The existence, uniqueness and convergence of formal Puiseux series solutions of non-autonomous algebraic differential equations of first order at a nonsingular point of the equation is studied, including the case where the celebrated…
Binary symmetry constraints are applied to constructing B\"acklund transformations of soliton systems, both continuous and discrete. Construction of solutions to soliton systems is split into finding solutions to lower-dimensional Liouville…
The discrete gradient approach is generalized to yield integral preserving methods for differential equations in Lie groups.
In this paper we study the construction of a discrete solution for a hyperbolic system of partial differentials of the strongly coupled type. In its construction, the discrete separation of matricial variable method was followed. Two…
Singularity subtraction for linear weakly singular Fredholm integral equations of the second kind is generalized to nonlinear integral equations. Two approaches are presented: The Classical Approach discretizes the nonlinear problem, and…
We study the adapted solution, numerical methods, and related convergence analysis for a unified backward stochastic partial differential equation (B-SPDE). The equation is vector-valued, whose drift and diffusion coefficients may involve…
An efficient linear solver plays an important role while solving partial differential equations (PDEs) and partial integro-differential equations (PIDEs) type mathematical models. In most cases, the efficiency depends on the stability and…
It is well known that isomonodromic deformations admit a Hamiltonian description. These Hamiltonians appear as coefficients of the characteristic equations of their Lax matrices, which define spectral curves for linear systems of…
The Hirota-Miwa equation can be written in `nonlinear' form in two ways: the discrete KP equation and, by using a compatible continuous variable, the discrete potential KP equation. For both systems, we consider the Darboux and binary…
An overview is given on recent developments in the affine Weyl group approach to Painlev\'e equations and discrete Painlev\'e equations, based on the joint work with Y. Yamada and K. Kajiwara.
We present a systematic and quite elementary method for constructing discrete Painlev\'e equations in the degeneration cascade for E$_8^{(1)}$. Starting from the invariant for the autonomous limit of the E$_8^{(1)}$ equation one wishes to…
A unique analytic continuation result is proven for solutions of a relatively general class of difference equations by using techniques of generalized Borel summability. We overview applications exponential asymptotics and analyzable…
We consider several examples of nonautonomous systems of difference equations coming from semi-classical orthogonal polynomials via recurrence coefficients and ladder operators, with respect to various generalisations of Laguerre and…
In this letter, the integrability aspects of a generalized Fisher type equation with modified diffusion in (1+1) and (2+1) dimensions are studied by carrying out a singularity structure and symmetry analysis. It is shown that the Painlev\'e…
Bilinear structure for the discrete Painlev\'e I equation is investigated. The solution on semi-infinite lattice is given in terms of the Casorati determinant of discrete Airy function. Based on this fact, the discrete Painlev\'e I equation…
The set of common numerical and analytical problems is introduced in the form of the generalized multidimensional discrete Poisson equation. It is shown that its solutions with square-summable discrete derivatives are unique up to a…
This paper applies methods of Van der Put and Van derPut-Saito to the fourth Painlev\'e equation. One obtains a Riemann--Hilbert correspondence between moduli spaces of rank two connections on $\mathbb{P}^1$ and moduli spaces for the…
We apply the Painleve test for integrability to a new discrete (differential-difference) nonlinear Schrodinger equation introduced by Leon and Manna. Since the singular expansions of solutions of this equation turn out to contain…
We prove a number of double-sided estimates relating discrete counterparts of several classical conformal invariants of a quadrilateral: cross-ratios, extremal lengths and random walk partition functions. The results hold true for any…
In an exploration paper, {\it L. Chen, Algorithms for Deforming and Contracting Simply Connected Discrete Closed Manifolds (I)}, we designed algorithms for deforming and contracting a simply connected discrete closed manifold to a discrete…