Deautonomization by singularity confinement: an algebro-geometric justification

Mase, Takafumi; Willox, Ralph; Grammaticos, B.; Ramani, A.

doi:10.1098/rspa.2014.0956

Cited by 20 publications

(64 citation statements)

References 14 publications

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“…In its original form, this method has the drawback that many systems with confined singularities have positive algebraic entropy [15], but recently singularity confinement has been refined to include information about deautonomization, which renders it a more effective tool [10,31,35]. If we apply the basic singularity confinement test to (2) for a few small values of N, then in all cases we find that the singularity pattern is…”

Section: Singularity Confinement and Laurentificationmentioning

confidence: 99%

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Some integrable maps and their Hirota bilinear forms

Hone

Kouloukas

Quispel

2017

J. Phys. A: Math. Theor.

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We introduce a two-parameter family of birational maps, which reduces to a family previously found by Demskoi, Tran, van der Kamp and Quispel (DTKQ) when one of the parameters is set to zero. The study of the singularity confinement pattern for these maps leads to the introduction of a tau function satisfying a homogeneous recurrence which has the Laurent property, and the tropical (or ultradiscrete) analogue of this homogeneous recurrence confirms the quadratic degree growth found empirically by Demskoi et al. We prove that the tau function also satisfies two different bilinear equations, each of which is a reduction of the Hirota-Miwa equation (also known as the discrete KP equation, or the octahedron recurrence). Furthermore, these bilinear equations are related to reductions of particular two-dimensional integrable lattice equations, of discrete KdV or discrete Toda type. These connections, as well as the cluster algebra structure of the bilinear equations, allow a direct construction of Poisson brackets, Lax pairs and first integrals for the birational maps. As a consequence of the latter results, we show how each member of the family can be lifted to a system that is integrable in the Liouville sense, clarifying observations made previously in the original DTKQ case.

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Section: Singularity Confinement and Laurentificationmentioning

confidence: 99%

“…Hence the Liouville integrability of the map (19) follows from that of (28), as described by the following result. (19) and (28) respectively, and let ψ denote the lift of (31) to N + 2 dimensions given by…”

Section: U-systems and Other Liouville Integrable Mapsmentioning

confidence: 99%

Some integrable maps and their Hirota bilinear forms

Hone

Kouloukas

Quispel

2017

J. Phys. A: Math. Theor.

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“…In this paper we shall reconsider the trustworthiness of singularity confinement as an integrability criterion. Based on recent findings which link the behaviour of the solutions of a mapping to that of its coefficients [4], we shall show that when one considers the non-autonomous extensions of mappings with confined singularities, one obtains a clear indication of the integrability or non-integrability of the mapping at hand and, in fact, that one can even compute its algebraic entropy exactly.…”

mentioning

confidence: 99%

“…We shall give several examples of this approach, concentrating on mappings that are not of QRT type, as these fall outside the class of mappings studied in [4].…”

mentioning

confidence: 99%

“…In [4] various non-autonomous mappings obtained from so-called late confinement -in which the singularities are not required to be confined at the earliest possible stage but only at some subsequent opportunity -were studied using algebro-geometric techniques. In particular, it was shown that for each of these mappings, the constraints on their parameters, as obtained from singularity confinement, were in fact equivalent to the linear transformation induced, by the mapping, on part of the Picard group of the (family of) rational surfaces on which it can be regularised by blowing-up from P 1 × P 1 .…”

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The redemption of singularity confinement

Ramani¹,

Grammaticos²,

Willox³

et al. 2015

J. Phys. A: Math. Theor.

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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.

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Singularity confinement and full-deautonomisation: A discrete integrability criterion

Grammaticos¹,

Ramani

Willox

et al. 2015

Physica D: Nonlinear Phenomena

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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.

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Deautonomization by singularity confinement: an algebro-geometric justification

Cited by 20 publications

References 14 publications

Some integrable maps and their Hirota bilinear forms

Some integrable maps and their Hirota bilinear forms

The redemption of singularity confinement

Singularity confinement and full-deautonomisation: A discrete integrability criterion

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