Takafumi Mase

Irreducibility and co-primeness as an integrability criterion for discrete equations

We study the Laurent property, the irreducibility and co-primeness of discrete integrable and non-integrable equations. First we study a discrete integrable equation related to the Somos-4 sequence, and also a non-integrable equation as a comparison. We prove that the conditions of irreducibility and co-primeness hold only in the integrable case. Next, we generalize our previous results on the singularities of the discrete Korteweg–de Vries (dKdV) equation. In our previous paper (Kanki et al 2014 J. Phys. A: Math. Theor. 47 065201) we described the singularity confinement test (one of the integrability criteria) using the Laurent property, and the irreducibility, and co-primeness of the terms in the bilinear dKdV equation, in which we only considered simplified boundary conditions. This restriction was needed to obtain simple (monomial) relations between the bilinear form and the nonlinear form of the dKdV equation. In this paper, we prove the co-primeness of the terms in the nonlinear dKdV equation for general initial conditions and boundary conditions, by using the localization of Laurent rings and the interchange of the axes. We assert that co-primeness of the terms can be used as a new integrability criterion, which is a mathematical re-interpretation of the confinement of singularities in the case of discrete equations.

Investigation into the role of the Laurent property in integrability

We study the Laurent property for autonomous and nonautonomous discrete equations. First we show, without relying on the caterpillar lemma, the Laurent property for the Hirota-Miwa and the discrete BKP (or so-called Miwa) equations. Next we introduce the notion of reductions and gauge transformations for discrete bilinear equations and we prove that these preserve the Laurent property. Using these two techniques, we obtain the explicit condition on the coefficients of a nonautonomous discrete bilinear equation for it to possess the Laurent property. Finally, we study the denominators of the iterates of an equation with the Laurent property and we show that any reduction to a mapping on a one-dimensional lattice of a nonautonomous Hirota-Miwa equation or discrete BKP equation, with the Laurent property, has zero algebraic entropy.

Deautonomization by singularity confinement: an algebro-geometric justification

The ‘deautonomization’ of an integrable mapping of the plane consists in treating the free parameters in the mapping as functions of the independent variable, the precise expressions of which are to be determined with the help of a suitable criterion for integrability. Standard practice is to use the singularity confinement criterion and to require that singularities be confined at the very first opportunity. An algebro-geometrical analysis will show that confinement at a later stage leads to a non-integrable deautonomized system, thus justifying the standard singularity confinement approach. In particular, it will be shown on some selected examples of discrete Painlevé equations, how their regularization through blow-up yields exactly the same conditions on the parameters in the mapping as the singularity confinement criterion. Moreover, for all these examples, it will be shown that the conditions on the parameters are in fact equivalent to a linear transformation on part of the Picard group, obtained from the blow-up.

The redemption of singularity confinement

We present a novel way to apply the singularity confinement property as a discrete integrability criterion. We shall use what we call a full deautonomisation approach, which consists in treating the free parameters in the mapping as functions of the independent variable, applied to a mapping complemented with terms that are absent in the original mapping but which do not change the singularity structure. We shall show, on a host of examples including the well-known mapping of Hietarinta-Viallet, that our approach offers a way to compute the algebraic entropy for these mappings exactly, thereby allowing one to distinguish between the integrable and non-integrable cases even when both have confined singularities.

Singularity confinement and full-deautonomisation: A discrete integrability criterion

Abstract We present a new approach to singularity confinement which makes it an efficient and reliable discrete integrability detector. Our method is based on the full-deautonomisation procedure, which consists in analysing non-autonomous extensions of a given discrete system obtained by adding terms that are initially absent, but whose presence does not alter the singularity pattern. A justification for this approach is given through an algebro-geometric analysis. We also introduce the notions of early and late confinement. While the former is a confinement that may exist already for the autonomous system, the latter corresponds to a singularity pattern longer than that of the autonomous case. Late confinement will be shown to play an important role in the singularity analysis of systems with non-trivial gauge freedom, for which the existence of an undetected gauge in conjunction with a sketchy analysis, might lead to erroneous conclusions as to their integrability. An algebro-geometric analysis of the role of late confinement in this context is also offered. This novel type of singularity confinement analysis will be shown to allow for the exact calculation of the algebraic entropy of a given mapping.

Algebraic entropy of an extended Hietarinta–Viallet equation

We introduce a series of discrete mappings, which is considered to be an extension of the Hietarinta-Viallet mapping with one parameter. We obtain the algebraic entropy for this mapping by obtaining the recurrence relation for the degrees of the iterated mapping. For some parameter values the mapping has a confined singularity, in which case the mapping is equivalent to a recurrence relation between irreducible polynomials. For other parameter values, the mapping does not pass the singularity confinement test. The properties of irreducibility and co-primeness of the terms play crucial roles in the discussion.

Coprimeness-preserving non-integrable extension to the two-dimensional discrete Toda lattice equation

We introduce a so-called coprimeness-preserving non-integrable extension to the two-dimensional Toda lattice equation. We believe that this equation is the first example of such discrete equations defined over a three-dimensional lattice. We prove that all the iterates of the equation are irreducible Laurent polynomials of the initial data and that every pair of two iterates is co-prime, which indicate confined singularities of the equation. By reducing the equation to two- or one-dimensional lattices, we obtain coprimeness-preserving non-integrable extensions to the one-dimensional Toda lattice equation and the Somos-4 recurrence.

Singularity confinement and chaos in two-dimensional discrete systems

We present a quasi-integrable two-dimensional lattice equation: i.e., a partial difference equation which satisfies a criterion of integrability, singularity confinement, although it has a chaotic aspect in the sense that the degrees of its iterates exhibit exponential growth. By systematic reduction to one-dimensional systems, it gives a hierarchy of ordinary difference equations with confined singularities, but with positive algebraic entropy including a generalized form of the Hietarinta-Viallet mapping. We believe that this is the first example of such quasi-integrable equations defined over a two-dimensional lattice.

Toda type equations over multi-dimensional lattices

We introduce a class of recursions defined over the

A two-dimensional lattice equation as an extension of the Heideman-Hogan recurrence

d