On Rational Solutions of Dressing Chains of Even Periodicity

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 $A_{N-1}^{(1)}$ invariant Painlevé 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 Painlevé 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.

arXiv comment29 pages

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Cited by in corpus (3)

• Rational Solutions of the Fifth Painlevé Equation. Generalised Laguerre Polynomials
• Two-fold degeneracy of a class of rational Painlevé V solutions
• Extended symmetry of higher Painlevé equations of even periodicity and their rational solutions