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arXiv · 2206.06482

On Rational Solutions of Dressing Chains of Even Periodicity

H. Aratyn·J. F. Gomes·G. V. Lobo·A. H. Zimerman
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Abstract

We develop a systematic approach to deriving rational solutions and obtaining classification of their parameters for dressing chains of even N periodicity or equivalently $A^{(1)}_{N-1}$ invariant Painlev\'e equations. This construction identifies rational solutions with points on orbits of fundamental shift operators acting on first-order polynomial solutions derived for dressing chains of even periodicity. We also obtain conditions for the existence of special function solutions that occur for a special class of first-order polynomial solutions. For the special case of the N=4 dressing chain equations the method yields all the known rational solutions of Painlev\'e V equation. They are obtained through action of shift operators on the two independent first-order polynomial solutions. The formalism naturally extends to N=6 and beyond as shown in the paper.

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H. Aratyn, J. F. Gomes, G. V. Lobo, A. H. Zimerman. 2022-06-13. On Rational Solutions of Dressing Chains of Even Periodicity. https://doi.org/10.3390/sym15010249

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