Singularities of the discrete KdV equation and the Laurent property

Abstract

We study the distribution of singularities for partial difference equations, in particular, the bilinear and nonlinear form of the discrete version of the Korteweg–de Vries (dKdV) equation. Using the Laurent property, and the irreducibility, and co-primeness of the terms of the bilinear dKdV equation, we clarify the relationship of these properties with the appearance of zeros in the time evolution. The results are applied to the nonlinear dKdV equation and we formulate the famous integrability criterion (singularity confinement test) for nonlinear partial difference equations with respect to the co-primeness of the terms.

Integrability criterion in terms of coprime property for the discrete Toda equation

“…Our results, along with preceding results for the discrete KdV equation and the Quispel-Roberts-Thompson type mappings [10,11], justify our assertion that the coprime property is an integrability detector. Since our results include the case of the equation with periodic boundary condition, which cannot be easily dealt with the singularity confinement approach, the coprime property is expected to be applicable to wider class of integrable and non-integrable mappings under various conditions than conventional integrability tests.…”

Studies on spaces of initial conditions for non-autonomous mappings of the plane

“…This approach is particularly useful when we numerically estimate the value of the entropy. The coprimeness condition was proposed to reinterpret singularity confinement from an algebraic viewpoint [22,24,23]. This criterion focuses on the factorization of iterates as rational functions of the initial values and tries to transform the equation to another one with the Laurent property [12].…”

Singularity confinement and chaos in two-dimensional discrete systems

“…Recently an algebraic reinterpretation of singularity confinement, co-primeness, was proposed so that it can apply to higher dimensional systems [20,21]. For a second order rational mapping, (x n , x n−1 ) → (x n+1 , x n ), x n is regarded as a rational function of the initial data (x 0 , x 1 ).…”