Ralph Willox

We examine the Lyness mapping (an integrable N th-order discrete system which can be generated from a one-dimensional reduction of the Hirota–Miwa equation) from the point of view of deautonomisation. We show that only the N = 2 case can be deautonomised when one works with the standard form of the mapping. However it turns out that deautonomisation is possible for arbitrary N when one considers the derivative form of the Lyness mapping. The deautonomisation of the derivative of the N = 2 case leads to a result we have never met before: the secular dependence in the coefficients of the mapping enters through two different exponential terms instead of just a single one. As a consequence, it turns out that a limit of this multiplicative dependence towards an additive one is possible without modifying the dependent variable. Finally, the analysis of the ‘late’ singularity confinement of the N = 2 case leads to a novel realisation of the full-deautonomisation principle: the dynamical degree is not given (as is customary) simply by the solution of some linear or multiplicative equation, but is present in the growth of the non-linear (and non-integrable) late-confinement conditions.

We derive a class of discrete Painlev\'e equations associated with the affine Weyl group E$_7^{(1)}$. The method used is the deautonomisation of a QRT mapping belonging to the canonical form VI (according to the classification of said mappings). An equation of such a form was the first instance of a symmetric -- in QRT parlance -- discrete analogue of the Painlev\'e VI equation. In this paper we present an exhaustive derivation of all the discrete Painlev\'e equations of this class. This is made possible thanks to previous studies that established the proper lengths of singularity patterns that are compatible with integrablity, and which were already successfully applied to the study of discrete Painlev\'e equations associated to the affine Weyl group E$_8^{(1)}$. Given that, from the latter, one can obtain by degeneration the equations related to E$_7^{(1)}$, we decided to link the results of the present study to those of the aforementioned ones. It turns out that a bridge from E$_8^{(1)}$ to E$_7^{(1)}$ exists in almost all cases, with one exception where, while in the former case a discrete Painlev\'e equation does exist, in the latter we find a mapping with only periodic coefficients, devoid of secular dependence.

The symmetry group of a (discrete) Painlevé equation provides crucial information on the properties of the equation. In this paper, we argue against the commonly held belief that the symmetry group of a given equation is solely determined by its surface type as given in the famous Sakai classification. We will dispel this misconception by using a specific example of a d-P_II equation, which corresponds to a half-translation on the root lattice dual to its surface-type root lattice but becomes a genuine translation on a sub-lattice thereof that corresponds to its real symmetry group. The latter fact is shown in two different ways, first by a brute force calculation, and then through the use of normalizer theory, which we believe to be an extremely useful tool for this purpose. We finish the paper with the analysis of a sub-case of our main example, which arises in the study of gap probabilities for Freud unitary ensembles, and the symmetry group of which is even further restricted due to the appearance of a nodal curve on the surface on which the equation is regularized.

We examine a collection of integrable and non-integrable birational mappings of order higher than two and show that the methods developed for calculating the degree growth of second order mappings can be transposed to higher-order systems. In particular, Halburd’s method which allows one to obtain explicitly the degree growth of a second order mapping, based on the patterns of its singularities, will be shown to work perfectly well in the higher-order case. The only precaution one must take is to choose initial conditions that are specifically adapted to the method. In the case of non-integrable mappings, one can apply either the express variant of Halburd’s method or the full-deautonomisation approach and we show that both lead to the (same) exact expression for the dynamical degree of the mapping.

The symmetry group of a (discrete) Painlevé equation provides crucial information on the properties of the equation. In this paper we argue against the commonly-held belief that the symmetry group of a given equation is solely determined by its surface type as given in the famous Sakai classification. We will dispel this misconception on a specific example of a d-${\text{P}_{\mathrm{II}}}$ equation which corresponds to a half-translation on the root lattice dual to its surface-type root lattice, but which becomes a genuine translation on a sub-lattice thereof that corresponds to its real symmetry group. The latter fact is shown in two different ways: first by a brute force calculation and second through the use of normalizer theory, which we believe to be an extremely useful tool for this purpose. We finish the paper with the analysis of a sub-case of our main example which arises in the study of gap probabilities for Freud unitary ensembles, and the symmetry group of which is even further restricted due to the appearance of a nodal curve on the surface on which the equation is regularized.

We present an algorithmic method for the calculation of the degrees of the iterates of birational mappings based on Halburd’s method for obtaining the degrees from the singularity structure of the mapping. The method uses only integer arithmetic with additions and, in some cases, multiplications by small integers. It is therefore extremely fast. Several examples of integrable and non-integrable mappings are presented. In the latter case, the dynamical degree we obtain from our method is always in agreement with that calculated by previously known methods.

We study the singularities of a modified lattice Korteweg–deVries (KdV) equation and show that it admits three families of singularities, with analogous properties to those found in the lattice KdV equation. The first family consists of localised singularities which can occupy an arbitrarily large domain but which are, nevertheless, always confined. The second family consists of one or more lines extending all the way from the South-West to the North-East on the plane, involving a single finite value that depends on the parameter that appears in the equation. We argue that the infinite extent of this singularity is not incompatible with the confinement property or with the integrability of the equation. The third family consists of horizontal strips in which the product of values on vertically adjacent lattice sites is equal to 1. In the case of the lattice KdV equation this type of singularity was dubbed ‘taishi’. The taishi for the modified lattice KdV equation can interact with singularities of the other two families, giving rise to very rich and quite intricate singularity structures. Nonetheless, these interactions can be described in a compact way through the formulation of a symbolic representation of the dynamics that is similar to, but in a sense, simpler than that for the KdV case. We give an interpretation of this symbolic representation in terms of a box & ball system related to the ultradiscrete modified KdV equation. These results show that taishi-type singularities are not limited to the lattice KdV equation, but might very well be a general feature of integrable lattice equations with deep connections to other facets of their integrability.