Coalescence, deformation and Bäcklund symmetries of Painlevé IV and II equations

Alves, V. C. C.; Aratyn, H.; Gomes, J. F.; Zímerman, A. H.

doi:10.1088/1751-8121/abb725

Cited by 3 publications

(10 citation statements)

References 24 publications

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“…with the transformed j i , α i , i = 1, 2, 3 now satisfying the conditions 3 i=1 j i = −3z/4, 3 i=1 α i = −3/2 that differ from the condition (3.4) by the minus sign. We can also run Kovalevskaya-Painlevé test as in references [21,3] by first assuming that solutions of (6.1) equations have to be of the form…”

Section: Discussionmentioning

confidence: 99%

“…In Section 6, that provides an outlook, we discuss the modification of the dressing chain that maintains the Kovalevskaya-Painlevé property together with the part of its Bäcklund symmetry. This relates to the program that we have pursued in several papers [3,4,5] attempting to connect integrability to the remaining symmetry of Painlevé models that survives the partial breaking of the extended affine Weyl group A (1) N −1 by deformation terms that preserve some notion of integrability.…”

Section: Introductionmentioning

confidence: 99%

“…This is the case of α ≡ N i=1 α i = 0 for which the periodic dressing chain admits a parametric dependent Lax representation [21]. Equation (3.6) with condition (3.9) for N = 3 yields I 30 (denoted as equation XXX in [15]) as shown in [3]. See reference [2] for relation between equation (3.6) with condition (3.9) for N = 4 and I 38 (equation XXXVIII in [15]).…”

Section: Introductionmentioning

confidence: 99%

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Gauge Symmetry Origin of Bäcklund Transformations for Painlevé Equations

Alves,

Aratyn,

Gomes

et al. 2021

Preprint

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We identify the self-similarity limit of the second flow of sl(N ) mKdV hierarchy with the periodic dressing chain thus establishing a connection to A(1)N −1 Bäcklund symmetries of dressing equations and Painlevé equations are obtained in the self-similarity limit of gauge transformations of the mKdV hierarchy realized as zero-curvature equations on the loop algebra sl(N ) endowed with a principal gradation.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Gauge Symmetry Origin of Bäcklund Transformations for Painlevé Equations

Alves,

Aratyn,

Gomes

et al. 2021

Preprint

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“…[9,12], and to expand equations with their possible degeneracy parameters allows one to obtain a hybrid equation with both a Painlevé equation and an elliptic equation, for example. This was done in [4], [3] and [2] and here the complete framework for such equations was provided since the basic recipe there was to first find the coalescence in the framework of symmetric equations and then extend the parameter space to all possible constants of integration the system provides. The coalescence cascades were seen here to be preserved for multiple properties and reductions, like for autonomous equations, Riccati equations, symmetric equations, and Bäcklund transformations.…”

Section: Discussionmentioning

confidence: 99%

“…Here we formulate coalescence in the setting of the symmetric P IV equations (64) through the following transformations [2] :…”

Section: Degeneracies On the Symmetric P IVmentioning

confidence: 99%

New Coalescences for the Painlevé Equations

Alves¹

2021

Preprint

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The Painlevé equations are here connected to other classes of equations with the Painlevé Property (Ince's equations) by the same degeneracy procedure that connects the Painlevé equations (coalescence). These Ince's equations here are also connected among themselves like in the traditional Painlevé's coalescence cascade. Such degeneracy is considered also for the special equations, symmetric equations and Bäcklund transformations.Composing these 24, we have the 6 Painlevé equations (P I , ..., P V I ), 6 autonomous equations solvable by Elliptic functions (I 3 , I 8 , I 12 , I 30 , I 38 , I 49 ), 7 absent of parameters (I 1 , I 2 , I 7 , I 11 , I 29 , I 32 , I 37 ), and 5 with arbitrary functions (I 5 , I 6 , I 14 , I 24 , I 27 ).P. Painlevé himself saw himself that it was possible to connect six Painlevé equations noticing that one can transform their variables and parameters artificially introducing a parameter (usually called ǫ) in a so specific way that the limiting procedure of ǫ → 0 turns an equation into some other. The P V I equation is considered a "master" equation for the other five since it degenerates into them [11].The Painlevé equations are non-linear differential equations with their critical points being just poles, they also have parameters that allow one to construct an infinite chain of solutions for each set of parameters through Bäcklund Transformations [10]. Since the degeneracy cascade mentioned above change the nature of the poles by "coalescing" them at each step, it is commonly known as coalescence cascade. The coalescence also coalesces the parameters as is seen in section (7) and in [14].The goal of this paper is to present a full degeneracy cascade connecting not only the usual 6 Painlevé equations, which is known but also the other 13 equations. The 5 equations with arbitrary functions, cannot be generated by this limiting procedure like the others, therefore will not be considered. Since the word "coalescence" brings an idea of quantities coalescing and this may not be the case in some situations here, I will keep it only for the Painlevé and autonomous equations, and calling by "degeneracy", which is more general, the other results presented here.

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Gauge symmetry origin of Bäcklund transformations for Painlevé equations

Alves¹,

Aratyn²,

Gomes³

et al. 2021

J. Phys. A: Math. Theor.

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We identify the self-similarity limit of the second flow of sl(N) mKdV hierarchy with the periodic dressing chain thus establishing a connection to A N − 1 ( 1 ) invariant Painlevé equations. The A N − 1 ( 1 ) Bäcklund symmetries of dressing equations and Painlevé equations are obtained in the self-similarity limit of gauge transformations of the mKdV hierarchy realized as zero-curvature equations on the loop algebra s l ̂ ( N ) endowed with a principal gradation.

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Coalescence, deformation and Bäcklund symmetries of Painlevé IV and II equations

Cited by 3 publications

References 24 publications

Gauge Symmetry Origin of Bäcklund Transformations for Painlevé Equations

Gauge Symmetry Origin of Bäcklund Transformations for Painlevé Equations

New Coalescences for the Painlevé Equations

Gauge symmetry origin of Bäcklund transformations for Painlevé equations

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