Related papers: Discrete Crum's Theorems and Integrable Lattice Eq…
The aim of this paper is the derivation of structure preserving schemes for the solution of the EPDiff equation, with particular emphasis on the two dimensional case. We develop three different schemes based on the Discrete Variational…
We introduce an efficient route to obtaining the discrete potential mKdV equation emerging from a particular discrete motion of discrete planar curves.
We present an approach to higher dimensional Darboux transformations suitable for application to quantum integrable systems and based on the bispectral property of partial differential operators. Specifically, working with the…
Various solutions to the discrete Schwarzian KdV equation are discussed. We first derive the bilinear difference equations of Hirota type of the discrete Schwarzian KP equation, which is decomposed into three discrete two-dimensional Toda…
We propose the systematic construction of classical and quantum two dimensional space-time lattices primarily based on algebraic considerations, i.e. on the existence of associated r-matrices and underlying spatial and temporal classical…
Miura-type transformations (MTs) are an essential tool in the theory of integrable nonlinear partial differential and difference equations. We present a geometric method to construct MTs for differential-difference (lattice) equations from…
We give a new mechanism for constructing Backlund transformations by using symmetry reduction of differential systems. We then characterize a family of Backlund transformations between Darboux integrable systems where the Backlund…
The purpose of this paper is to bridge the gap between the Dbar method and the direct linearization approach for the lattice Korteweg-de Vries (KdV) type equations. We develop the Dbar method to study some discrete integrable equations in…
Darboux transformation of a second-order linear differential operator is a well-known technique with many applications in mathematics and physics. We study Darboux transformation from the point of view of Markov semigroups of diffusion…
We investigate the relation between the local variables of a discrete integrable lattice system and the corresponding separation variables, derived from the associated spectral curve. In particular, we have shown how the inverse…
The action of a B\"acklund-Darboux transformation on a spectral problem associated with a known integrable system can define a new discrete spectral problem. In this paper, we interpret a slightly generalized version of the binary…
We propose a systematic method for constructing integrable delay-difference and delay-differential analogues of known soliton equations such as the Lotka-Volterra, Toda lattice, and sine-Gordon equations and their multi-soliton solutions.…
We consider here two discrete versions of the modified KdV equation. In one case, some solitary wave solutions, B\"acklund transformations and integrals of motion are known. In the other one, only solitary wave solutions were given, and we…
Integrable discrete scalar equations defined on a~two or a three dimensional lattice can be rewritten as difference systems in bond variables or in face variables respectively. Both the difference systems in bond variables and the…
We analyze the canonical treatment of classical constrained mechanical systems formulated with a discrete time. We prove that under very general conditions, it is possible to introduce nonsingular canonical transformations that preserve the…
In this paper, we propose a new approach to calculate multi-soliton solutions of Camassa-Holm (CH) equation and modified Camassa-Holm (MCH) equation with aid of Darboux transformation (DT). The new approach simplifies the approach presented…
A new approach for obtaining the transformations of solutions of nonlinear ordinary differential equations representable as the compatibility condition of the overdetermined linear systems is proposed. The corresponding transformations of…
A $2n$-dimensional Lax integrable system is proposed by a set of specific spectral problems. It contains Takasaki equations, the self-dual Yang-Mills equations and its integrable hierarchy as examples. An explicit formulation of Darboux…
In this work, following the discrete de Rham (DDR) approach, we develop a discrete counterpart of a two-dimensional de Rham complex with enhanced regularity. The proposed construction supports general polygonal meshes and arbitrary…
A method of integrable discretization of the Liouville type nonlinear partial differential equations is suggested based on integrals. New examples of discrete Liouville type models are presented.