Special Polynomials Associated with Rational and Algebraic Solutions of the Painlevé Equations

نویسندگان

  • Peter A. Clarkson
  • P. A. CLARKSON
چکیده

— Rational solutions of the second, third and fourth Painlevé equations (PII–PIV) can be expressed in terms of logarithmic derivatives of special polynomials that are defined through coupled second order, bilinear differential-difference equations which are equivalent to the Toda equation. In this paper the structure of the roots of these special polynomials, and the special polynomials associated with algebraic solutions of the third and fifth Painlevé equations, is studied and it is shown that these have an intriguing, highly symmetric and regular structure. Further, using the Hamiltonian theory for PII–PIV, it is shown that all these special polynomials, which are defined by differential-difference equations, also satisfy fourth order, bilinear ordinary differential equations. Résumé (Polynômes spéciaux associés aux solutions rationnelles ou algébriques des équations de Painlevé) On peut exprimer les solutions rationnelles des équations PII, PIII et PIV en fonction des dérivées logarithmiques de polynômes spéciaux définis par des équations différences-différentielles bilinéaires d’ordre deux couplées et équivalentes à l’équation de Toda. Dans cet article nous étudions la configuration des racines de ces polynômes spéciaux et des polynômes spéciaux associés aux solutions algébriques des équations de Painlevé PIII et PV. Nous mettons en évidence une structure étonnante, fortement symétrique et régulière. En outre, appliquant la théorie hamiltonienne à PII, PIII et PIV, nous montrons que tous ces polynômes spéciaux, définis par des équations différences-différentielles, satisfont aussi à des équations différentielles ordinaires bilinéaires d’ordre 4. 2000 Mathematics Subject Classification. — 33E17, 34M35.

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تاریخ انتشار 2007