Related papers: On construction of discrete Darboux integrable equ…
Characteristic Lie algebras of semi-discrete chains are studied. The attempt to adopt this notion to the classification of Darboux integrable chains has been undertaken.
In this paper we shall study differential equations in the complex domain. The method of indeterminate coefficients and the majorant method lead to a proof of the existence and uniqueness of meromorphic solution of differential equations.…
We extend two of the methods previously introduced to find discrete symmetries of differential equations to the case of difference and differential-difference equations. As an example of the application of the methods, we construct the…
A discretisation scheme that preserves topological features of a physical problem is extended so that differential geometric structures can be approximated in a consistent way thus giving access to the study of physical systems which are…
The article studies a class of integrable semidiscrete equations with one continuous and two discrete independent variables. Miura type transformations are obtained that relate the equations of the class. A new integrable chain of this type…
New extensions of the KP and modified KP hierarchies with self-consistent sources are proposed. The latter provide new generalizations of $(2+1)$-dimensional integrable equations, including the DS-III equation and the $N$-wave problem.…
Darboux transformations in one independent variable have found numerous applications in various field of mathematics and physics. In this paper we show that the extension of these transformations to two dimensions can be used to decouple…
In this paper we analyze the tangential symmetries of Darboux integrable decomposable exterior differential systems. The decomposable systems generalize the notion of a hyperbolic exterior differential system and include the classic notion…
In the article the problem of the integrable classification of nonlinear lattices depending on one discrete and two continuous variables is studied. By integrability we mean the presence of reductions of a chain to a system of hyperbolic…
A method is presented to obtain the change in the potential and in the relevant wavefunction of a linear system of ordinary differential equations containing a spectral parameter, when that linear system is perturbed and a finite number of…
Integrable discretizations of the complex and real Dym equations are proposed. N-soliton solutions for both semi-discrete and fully discrete analogues of the complex and real Dym equations are also presented.
The paper is devoted to the conjecture that an equation is Darboux integrable if and only if it possesses symmetries depending on arbitrary functions. We note that results of previous works together prove this conjecture for scalar partial…
We present the Darboux transformations for a novel class of two-dimensional discrete integrable systems named as $\mathbb{Z}_\mathcal{N}$ graded discrete integrable systems, which were firstly proposed by Fordy and Xenitidis within the…
We investigate solvability of a continuous Dirichlet boundary value problem together with its classical discretization using a gobal diffeomorphism theorem.
Following the usual definition of $\lambda$-symmetries of differential equations, we introduce the analogous concept for difference equations and apply it to some examples.
The autonomous Hietarinta equation is a well-known example of the quad-graph discrete equation which is consistent around the cube. In a recent work, it was conjectured that this equation is Darboux integrable (i.e., for each of two…
A new class of integrable maps, obtained as lattice versions of polynomial dynamical systems is introduced. These systems are obtained by means of a discretization procedure that preserves several analytic and algebraic properties of a…
We study discrete isospectral symmetries for the classical acoustic spectral problem in spatial dimensions one and two, by developing a Darboux (Moutard) transformation formalism for this problem. The procedure follows the steps, similar to…
We consider delay differential equations with a polynomially distributed delay. We derive an equivalent system of delay differential equations, which includes just two discrete delays. The stability of the equivalent system and its…
We express Darboux transformations of discrete polarised curves as parallel sections of discrete connections in the quaternionic formalism. This immediately leads to the linearisation of the monodromy of the transformation. We also consider…