Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 008, 27 pages      arXiv:2308.16051      https://doi.org/10.3842/SIGMA.2024.008
Contribution to the Special Issue on Evolution Equations, Exactly Solvable Models and Random Matrices in honor of Alexander Its' 70th birthday

Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III $({\rm D}_7)$ Equation

Robert J. Buckingham ^a and Peter D. Miller ^b
a) Department of Mathematical Sciences, University of Cincinnati, P.O. Box 210025, Cincinnati, OH 45221, USA
^b) Department of Mathematics, University of Michigan, East Hall, 530 Church St., Ann Arbor, MI 48109, USA

Received August 31, 2023, in final form January 05, 2024; Published online January 20, 2024

Abstract
It is well known that the Painlevé equations can formally degenerate to autonomous differential equations with elliptic function solutions in suitable scaling limits. A way to make this degeneration rigorous is to apply Deift-Zhou steepest-descent techniques to a Riemann-Hilbert representation of a family of solutions. This method leads to an explicit approximation formula in terms of theta functions and related algebro-geometric ingredients that is difficult to directly link to the expected limiting differential equation. However, the approximation arises from an outer parametrix that satisfies relatively simple conditions. By applying a method that we learned from Alexander Its, it is possible to use these simple conditions to directly obtain the limiting differential equation, bypassing the details of the algebro-geometric solution of the outer parametrix problem. In this paper, we illustrate the use of this method to relate an approximation of the algebraic solutions of the Painlevé-III (D$_7$) equation valid in the part of the complex plane where the poles and zeros of the solutions asymptotically reside to a form of the Weierstraß equation.

Key words: Painlevé-III (D$_7$) equation; isomonodromy method; algebraic solutions; Weierstraß equation.

pdf (5492 kb)   tex (5584 kb)  

References

1. Bothner T., Miller P.D., Rational solutions of the Painlevé-III equation: large parameter asymptotics, Constr. Approx. 51 (2020), 123-224, arXiv:1808.01421.
2. Buckingham R.J., Miller P.D., Large-degree asymptotics of rational Painlevé-II functions: noncritical behaviour, Nonlinearity 27 (2014), 2489-2578, arXiv:1310.2276.
3. Buckingham R.J., Miller P.D., Large-degree asymptotics of rational Painlevé-IV solutions by the isomonodromy method, Constr. Approx. 56 (2022), 233-443, arXiv:2008.00600.
4. Buckingham R.J., Miller P.D., On the algebraic solutions of the Painlevé-III $(\rm D_7)$ equation, Phys. D 441 (2022), 133493, 22 pages, arXiv:2202.04217.
5. Clarkson P.A., The third Painlevé equation and associated special polynomials, J. Phys. A 36 (2003), 9507-9532.
6. Deift P., Zhou X., A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. 137 (1993), 295-368.
7. Dubrovin B.A., Theta-functions and nonlinear equations, Russian Math. Surveys 36 (1981), 11-92.
8. Fokas A.S., Its A.R., Kapaev A.A., Novokshenov V.Yu., Painlevé transcendents: The Riemann-Hilbert approach, Math. Surveys Monogr., Vol. 128, American Mathematical Society, Providence, RI, 2006.
9. Its A.R., Kapaev A.A., The nonlinear steepest descent approach to the asymptotics of the second Painlevé transcendent in the complex domain, in MathPhys Odyssey, 2001, Prog. Math. Phys., Vol. 23, Birkhäuser, Boston, MA, 2002, 273-311, arXiv:nlin/0108054.
10. Jenkins J.A., Univalent functions and conformal mapping, Ergeb. Math. Grenzgeb. (3), Vol. 18, Springer, Berlin, 1958.
11. Kitaev A.V., Vartanian A.H., Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I, Inverse Problems 20 (2004), 1165-1206, arXiv:math.CA/0312075.
12. Ohyama Y., Kawamuko H., Sakai H., Okamoto K., Studies on the Painlevé equations. V. Third Painlevé equations of special type ${\rm P}_{\rm III}({\rm D}_7)$ and ${\rm P}_{\rm III}({\rm D}_8)$, J. Math. Sci. Univ. Tokyo 13 (2006), 145-204.
13. Olver F.W.J., Olde Daalhuis A.B., Lozier D.W., Schneider B.I., Boisvert R.F., Clark C.W., Miller B.R., Saunders B.V., Cohl H.S., McClain M.A., NIST digital library of mathematical functions, Release 1.1.10 of 2023-06-15, aviable at https://dlmf.nist.gov/.
14. Shimomura S., Boutroux ansatz for the degenerate third Painlevé transcendents, arXiv:2207.11495.
15. Strebel K., Quadratic differentials, Ergeb. Math. Grenzgeb. (3), Vol. 5, Springer, Berlin, 1984.