Lax pairs for ultra-discrete Painlevé cellular automata

Joshi, Nalini; Nijhoff, F.W.; Ormerod, Chris

doi:10.1088/0305-4470/37/44/l03

Cited by 16 publications

(26 citation statements)

References 9 publications

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“…Such equations are naturally expressed in terms of the max-plus semi-ring and arise through the ultra-discretization of known integrable discrete equations [28,18]. There has been particular interest in the ultra-discrete Painlevé equations [27,22,5,15], which are tropical versions of the discrete Painlevé equations. Joshi and Lafortune [13] have described an analogue of singularity confinement for ultra-discrete equations.…”

Section: Introductionmentioning

confidence: 99%

Tropical Nevanlinna Theory and Ultradiscrete Equations

Halburd

Southall

2008

International Mathematics Research Notices

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A tropical version of Nevanlinna theory is described in which the role of meromorphic functions is played by continuous piecewise linear functions of a real variable whose one-sided derivatives are integers at every point. These functions are naturally defined on the max-plus (or tropical) semi-ring. Analogues of the Nevanlinna characteristic, proximity and counting functions are defined and versions of Nevanlinna's first main theorem, the lemma on the logarithmic derivative and Clunie's lemma are proved.As well as providing another example of a tropical or dequantized analogue of an important area of complex analysis, this theory has applications to so-called ultra-discrete equations. Preliminary results are presented suggesting that the existence of finite-order max-plus meromorphic solutions can be considered to be an ultra-discrete analogue of the Painlevé property.

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Section: Introductionmentioning

confidence: 99%

Tropical Nevanlinna Theory and Ultradiscrete Equations

Halburd

Southall

2008

International Mathematics Research Notices

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“…The resulting system may be defined over the integers, hence, these systems are often referred to as cellular automata [14].…”

Section: Ultradiscretizationmentioning

confidence: 99%

Tropical geometric interpretation of ultradiscrete singularity confinement

Ormerod¹

2013

J. Phys. A: Math. Theor.

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Abstract. Using the interpretation of the ultradiscretization procedure as a non-Archimedean valuation, we use results of tropical geometry to show how roots and poles manifest themselves in piece-wise linear systems as points of non-differentiability. This will allow us to demonstrate a correspondence between singularity confinement for discrete integrable systems and ultradiscrete singularity confinement for ultradiscrete integrable systems.

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“…The ultradiscrete Painlevé equations are second order non-linear difference equations defined over the max-plus semifield that are integrable in the sense that they possess many of same properties of the continuous and discrete Painlevé equations that are associated with integrability, albeit, in some tropical form. These properties include tropical Lax representations [15,35] and tropical singularity confinement [14,36]. They also admit symmetry groups of affine Weyl type [18,19] and special solutions of rational and hypergeometric type [26,34,50].…”

Section: Introductionmentioning

confidence: 99%

From Polygons to Ultradiscrete Painlevé Equations

Ormerod

Yamada

2015

SIGMA

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Abstract. The rays of tropical genus one curves are constrained in a way that defines a bounded polygon. When we relax this constraint, the resulting curves do not close, giving rise to a system of spiraling polygons. The piecewise linear transformations that preserve the forms of those rays form tropical rational presentations of groups of affine Weyl type. We present a selection of spiraling polygons with three to eleven sides whose groups of piecewise linear transformations coincide with the Bäcklund transformations and the evolution equations for the ultradiscrete Painlevé equations.

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Lax pairs for ultra-discrete Painlevé cellular automata

Cited by 16 publications

References 9 publications

Tropical Nevanlinna Theory and Ultradiscrete Equations

Tropical Nevanlinna Theory and Ultradiscrete Equations

Tropical geometric interpretation of ultradiscrete singularity confinement

From Polygons to Ultradiscrete Painlevé Equations

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