相关论文: Darboux transformations for a 6-point scheme

With this paper we begin an investigation of difference schemes that possess Darboux transformations and can be regarded as natural discretizations of elliptic partial differential equations. We construct, in particular, the Darboux…

Darboux Transformation, well known in second order differential operator theory, is applied here to the difference equation satisfied by the discrete hypergeometric polynomials(Charlier, Meixner-Krawchuk, Hahn).

Darboux transformations for linear operators on regular two dimensional lattices are reviewed. The six point scheme is considered as the master linear problem, whose various specifications, reductions, and their sublattice combinations lead…

In this paper we implement the Darboux transformation, as well as an analogue of Crum's theorem, for a discrete version of Schr\"odinger equation. The technique is based on the use of first order operators intertwining two difference…

We have been working in many aspects of the problem of analyzing, understanding and solving ordinary differential equations (first and second order). As we have extensively mentioned, while working in the Darboux type methods, the most…

All Darboux integrable difference equations on the quad-graph are described in the case of the equations that possess autonomous first-order integrals in one of the characteristics. A generalization of the discrete Liouville equation is…

We prove that second-order hyperbolic Monge-Ampere equations for one function of two variables are connected to the wave equation by a Backlund transformation if and only if they are integrable by the method of Darboux at second order. One…

We present some basic properties of two distinguished discretizations of elliptic operators: the self-adjoint 5-point and 7-point schemes on a two dimensional lattice. We first show that they allow to solve Dirichlet boundary value…

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…

We present a general solution-generating result within the bidifferential calculus approach to integrable partial differential and difference equations, based on a binary Darboux-type transformation. This is then applied to the…

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.

We develop the Darboux procedure for the case of the two-level system. In particular, it is demonstrated that one can construct the Darboux intertwining operator that does not violate the specific structure of the equations of the two-level…

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…

Here, Darboux's classical results about transformations with differential substitutions for hyperbolic equations are extended to the case of parabolic equations of the form $L u = \big(D^2_{x} + a(x,y) D_x + b(x,y) D_y + c(x,y)\big)u=0$. We…

In this paper we present a far-reaching generalization of E. Vessiot's analysis of the Darboux integrable partial differential equations in one dependent and two independent variables. Our approach provides new insights into this classical…

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 second half of the 19th century Darboux obtained determinant formulae that provide the general solution for a linear hyperbolic second order PDE with finite Laplace series. These formulae played an important role in his study of the…

We obtain the well-known discrete modified Boussinesq equation in two-component form as well as its Lax pair in $3\times3$ matrix form through a 3-periodic reduction technique on the Hirota-Miwa equation and its Lax pair. We describe how…

The article investigates systems of differential-difference equations of hyperbolic type, integrable in sense of Darboux. The concept of a complete set of independent characteristic integrals underlying Darboux integrability is discussed. A…

We construct new quasi-exactly solvable one-dimensional potentials through Darboux transformations. Three directions are investigated: Reducible and two types of irreducible second-order transformations. The irreducible transformations of…