A Class of Integrable and Nonintegrable Mappings and their Dynamics

Tsuda, Teruhisa; Grammaticos, B.; Ramani, A.; Takenawa, Tomoyuki

doi:10.1007/s11005-007-0200-0

Cited by 31 publications

(55 citation statements)

References 13 publications

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“…7 . Thus, thanks to the chain of Miura transformations (42-43) and (47-48), together with the translation (45), we were able to bring an equation, associated with the group E It is interesting to point out here that the autonomous limit of equation (46), or equivalently (44), does not lead to a QRT-type mapping, but rather to a form first encountered in [19]. We find that the quantity…”

Section: Systemmentioning

confidence: 82%

“…a ratio of polynomials that are quadratic in y n and y n−1 . Thus we conclude that, starting from the HKY mapping for x with bi-quartic invariant (10), by combining the Miura transformations (14-15) and (18)(19), we can transform it into a mapping for y with an invariant of QRT type.…”

Section: Systemmentioning

confidence: 89%

“…Deautonomising it, we recognised an equation associated to the affine Weyl group E (1) 6 , already obtained in [11]. Again we considered the possible Miura transformations and while the equation is indeed of QRT type at the autonomous limit, the first Miura cast it in a non-QRT form, reminiscent of systems we studied in [19]. A final Miura transformation brought the equation to a form that is canonical for discrete Painlevé equations associated to the group E (1) 7 (and it is not clear whether a transformation to a canonical E (1) 6 form is possible).…”

Section: Discussionmentioning

confidence: 99%

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Miura transformations for discrete Painlevé equations coming from the affine E8 Weyl group

Ramani

Grammaticos

Willox

2017

Journal of Mathematical Physics

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We derive integrable equations starting from autonomous mappings with a general form inspired by the additive systems associated to the affine Weyl group E (1) 8 . By deautonomisation we obtain two hitherto unknown systems, one of which turns out to be a linearisable one, and we show that both these systems arise from the deautonomisation of a non-QRT mapping. In order to unambiguously prove the integrability of these nonautonomous systems, we introduce a series of Miura transformations which allows us to prove that one of these systems is indeed a discrete Painlevé equation, related to the affine Weyl group E (1) 7 , and to cast it in canonical form. A similar sequence of Miura transformations allows us to effectively linearise the second system we obtain. An interesting offshoot of our calculations is that the series of Miura transformations, when applied at the autonomous limit, allows one to transform a non-QRT invariant into a QRT one.

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Section: Systemmentioning

confidence: 82%

Section: Systemmentioning

confidence: 89%

Section: Discussionmentioning

confidence: 99%

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Miura transformations for discrete Painlevé equations coming from the affine E8 Weyl group

Ramani

Grammaticos

Willox

2017

Journal of Mathematical Physics

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“…The relation of HKY mappings to QRT ones through the general conservation law (1.12) we found in [5] is not an isolated occurence. Indeed, in [7] we obtained the mapping…”

Section: Folding Transformations and Hky Mappings 77mentioning

confidence: 99%

Folding Transformations and HKY Mappings

Grammaticos¹,

Ramani²,

Willox³

2021

JNMP

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We present a new method for the derivation of mappings of HKY type. These are second-order mappings which do not have a biquadratic invariant like the QRT mappings, but rather an invariant of degree higher than two in at least one of the variables. Our method is based on folding transformations which exist for some discrete Painlevé equations. They are transformations which relate the variable of a discrete Painlevé equation to the square of the variable of some other one. By considering the autonomous limit of these relations we derive folding-like transformations which relate QRT mappings to HKY ones. We construct the invariants of the latter mappings and show how they can be extended beyond the ones given by the strict application of the folding transformation.

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“…As shown in [15], two singularities exist when either x = 1 or x = a. The first singularity is confined leading to a finite singularity pattern {1, ∞, a}.…”

mentioning

confidence: 99%

Do All Integrable Evolution Equations Have the Painlevé Property?

Tamizhmani¹,

Grammaticos²,

Ramani³

et al. 2007

SIGMA

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Abstract. We examine whether the Painlevé property is necessary for the integrability of partial differential equations (PDEs). We show that in analogy to what happens in the case of ordinary differential equations (ODEs) there exists a class of PDEs, integrable through linearisation, which do not possess the Painlevé property. The same question is addressed in a discrete setting where we show that there exist linearisable lattice equations which do not possess the singularity confinement property (again in analogy to the one-dimensional case).

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A Class of Integrable and Nonintegrable Mappings and their Dynamics

Cited by 31 publications

References 13 publications

Miura transformations for discrete Painlevé equations coming from the affine E8 Weyl group

Miura transformations for discrete Painlevé equations coming from the affine E8 Weyl group

Folding Transformations and HKY Mappings

Do All Integrable Evolution Equations Have the Painlevé Property?

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