Discrete Korteweg-de Vries Equation Research Articles

Consistency around a cube property of Hirota's discrete KdV equation and the lattice sine-Gordon equation

It has been unknown whether Hirota's discrete Korteweg-de Vries equation and the lattice sine-Gordon equation have the consistency around a cube (CAC) property. In this paper, we show that they have the CAC property. Moreover, we also show that they can be extended to systems on the 3-dimensional integer lattice.

Bäcklund transformation and multi-soliton solutions for the discrete Korteweg–de Vries equation

In this paper, a discrete Korteweg–de Vries equation for an LC network consisting of the voltage-dependent capacitors and current-dependent inductors is investigated by symbolic computation. Binary Bell polynomials are applied to the discrete Korteweg–de Vries equation which is reduced to the corresponding bilinear form directly rather than transformed into its fourlinear one first. Simultaneously, the dependent variable transformation is acquired through the deriving procedures. Bäcklund transformation of the dKdV equation is derived with the introduced mixing variables instead of the exchange formulae. Solitonic propagation and interaction in the electrical circuit are illustrated and analyzed.

On the Darboux transform and the solutions of some integrable systems

The relation between the Darboux transformation and the solutions of the full Kostant Toda lattice is analyzed. The discrete Korteweg de Vries equation is used to obtain such solutions and the main result of [1] is extended to the case of general (p + 2)-banded Hessenberg matrices.

Twisted reductions of integrable lattice equations, and their Lax representations

It is well known that from two-dimensional lattice equations one can derive one-dimensional lattice equations by imposing periodicity in some direction. In this paper we generalize the periodicity condition by adding a symmetry transformation and apply this idea to autonomous and non-autonomous lattice equations. As results of this approach, we obtain new reductions of the discrete potential Korteweg–de Vries (KdV) equation, discrete modified KdV equation and the discrete Schwarzian KdV equation. We will also describe a direct method for obtaining Lax representations for the reduced equations.

Integrability of reductions of the discrete Korteweg–de Vries and potential Korteweg–de Vries equations

We study the integrability of mappings obtained as reductions of the discrete Korteweg–de Vries (KdV) equation and of two copies of the discrete potential KdV (pKdV) equation. We show that the mappings corresponding to the discrete KdV equation, which can be derived from the latter, are completely integrable in the Liouville–Arnold sense. The mappings associated with two copies of the pKdV equation are also shown to be integrable.

Discrete Painlevé equations and their Lax pairs as reductions of integrable lattice equations

We describe a method to obtain Lax pairs for periodic reductions of a rather general class of integrable non-autonomous lattice equations. The method is applied to obtain reductions of the non-autonomous discrete Korteweg–de Vries equation and non-autonomous discrete Schwarzian Korteweg–de Vries equation, which yield a discrete analogue of the fourth Painlevé equation, a q-analogue of the sixth Painlevé equation and the q-Painlevé equation with a symmetry group of affine Weyl type E(1)6.

On the full Kostant–Toda system and the discrete Korteweg–de Vries equations

The relation between the solutions of the full Kostant–Toda lattice and the discrete Korteweg–de Vries equation is analyzed. A method for constructing solutions of these systems is given. As a consequence of the matricial interpretation of this method, the transform of Darboux is extended for general Hessenberg banded matrices.

Infinitely many symmetries and conservation laws for quad-graph equations via the Gardner method

The application of the Gardner method for the generation of conservation laws to all the ABS equations is considered. It is shown that all the necessary information for the application of the Gardner method, namely Bäcklund transformations and initial conservation laws, follows from the multidimensional consistency of ABS equations. We also apply the Gardner method to an asymmetric equation which is not included in the ABS classification. An analog of the Gardner method for the generation of symmetries is developed and applied to the discrete Korteweg–de Vries equation. It can also be applied to all the other ABS equations.

On the shapes of two-dimensional soliton perturbations in simple lattices

The Toda lattice and the discrete Korteweg-de Vries equation generalized to two dimensions are studied numerically. The interactions are assumed to be identical in both directions. It is shown that the equations have solutions in the form of plane linear and localized solitons. In contrast to equations integrable by the inverse scattering method, the parameters of solitons change in the course of their interaction and additional wave structures are formed. The basic types of solutions characterizing these processes are presented.

Soliton solutions to generalized discrete Korteweg-de Vries equations

New discrete equations of the simplest three-point form are considered that generalize the discrete Korteweg-de Vries equation. The properties of solitons, kinks, and oscillatory waves are numerically examined for three types of interactions between neighboring chain elements. An analogy with solutions to limiting continual equations is drawn.

Discrete and ultradiscrete Bäcklund transformation for KdV equation

The Bäcklund transformation for the discrete Korteweg–de Vries equation is introduced in the bilinear form. The superposition formula is also derived from the transformation. An ultradiscrete analogue of the transformation is presented by means of the ultradiscretization technique. This analogue gives the Bäcklund transformation for the box and ball system. The ultradiscrete soliton solutions for the system are also discussed with explicit examples.

Nonlinear differential-difference and difference equations: integrability and exact solvability

A brief review on the recent results of nonlinear differential-difference and difference equations toward its complete integrability and exact solvability is presented. In particular, we show how Lie's theory of differential equations can be extended to differential-difference and pure difference equations and illustrate its applicability through the discrete Korteweg-deVries equation as an example. Also, we report that an autonomous nonlinear difference equation of an arbitrary order with one or more independent variables can be linearised by a point transformation if and only if it admits a symmetry vector field whose coefficient is the product of two functions, one of the dependent variable and of the independent variables. This is illustrated by linearising several first- and second-order ordinary nonlinear difference equations. A possible connection between the Lie symmetry analysis and the onset of chaos with reference to first-order mappings is explored.

Isothermic Surfaces in Spaces of Arbitrary Dimension: Integrability, Discretization, and Bäcklund Transformations—A Discrete Calapso Equation

A vector analog of the classical Calapso equation governing isothermic surfaces in Rn+2 is introduced. It is shown that this vector Calapso system admits a nonlocal) scalar Lax pair based on the classical Moutard equation. The analog of Darboux's Bäcklund transformation for isothermic surfaces in R3 is derived in a systematic manner and shown that it may be formulated in terms of the classical Moutard transformation acting on the scalar Lax pair. A permutability theorem for isothermic surfaces is set down that manifests itself in an explicit superposition principle for the vector Calapso system. This superposition principle in vectorial form is shown to constitute an integrable discretization of the vector Calapso system and, therefore, defines discrete isothermic surfaces in Rn+2. The discrete Calapso equation is related to the discrete Korteweg–de Vries equation and discrete holomorphic functions. A matrix Lax pair based on Clifford algebras and a scalar Lax pair are derived for the discrete Calapso equation. A discrete Moutard‐type transformation for the discrete Calapso equation is obtained, and it is shown that the discrete Calapso equation may be specialized to an integrable discrete version of the O(n+2) nonlinear σ‐model.

Superposition of solitons and discrete evolution equations

Solutions of differential-difference Korteweg-de Vries (KdV), modified KdV and generalized KdV equations are given in terms of theta functions. The dispersion relation is given in elegant, compact forms and the solutions consist of a sequence of solitions. A fully discrete or partial difference KdV equation is also treated. A new identity in theta function enables the elliptic function solution to be rewritten as a sum of solitons.