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

SIGMA 15 (2019), 046, 53 pages      arXiv:1809.00122      https://doi.org/10.3842/SIGMA.2019.046

Meromorphic Solution of the Degenerate Third Painlevé Equation Vanishing at the Origin

Alexander V. Kitaev
Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191023, Russia

Received November 13, 2018, in final form May 30, 2019; Published online June 18, 2019

Abstract
We prove that there exists the unique odd meromorphic solution of dP3, $u(\tau)$ such that $u(0)=0$, and study some of its properties, mainly: the coefficients of its Taylor expansion at the origin andasymptotic behaviour as $\tau\to+\infty$.

Key words: Painlevé equation; asymptotic expansion; hypergeometric function; isomonodromy deformation; greatest common divisor.

pdf (960 kb)   tex (306 kb)  

References

  1. Bilman D., Ling L., Miller P.D., Extreme superposition: roague waves of infinite order and the Painlevé-III hierarchy, arXiv:1806.00545.
  2. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. I, McGraw-Hill Book Company, Inc., New York - Toronto - London, 1953.
  3. Gamayun O., Iorgov N., Lisovyy O., How instanton combinatorics solves Painlevé VI, V and IIIs, J. Phys. A: Math. Theor. 46 (2013), 335203, 29 pages, arXiv:1302.1832.
  4. Garnier R., Sur des équations différentielles du troisième ordre dont l'intégrale générale est uniforme et sur une classe d'équations nouvelles d'ordre supérieur dont l'intégrale générale a ses points critiques fixes, Ann. Sci. École Norm. Sup. (3) 29 (1912), 1-126.
  5. Hardy G.H., Littlewood J.E., Tauberian theorems concerning power series and Dirichlet's series whose coefficients are positive, Proc. London Math. Soc. 13 (1914), 174-191.
  6. Hardy G.H., Wright E.M., An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979.
  7. Kitaev A.V., Vartanian A., Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I, Inverse Problems 20 (2004), 1165-1206, arXiv:math.CA/0312075.
  8. Kitaev A.V., Vartanian A., Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation: II, Inverse Problems 26 (2010), 105010, 58 pages, arXiv:1005.2677.
  9. Kitaev A.V., Vartanian A., Asymptotics of integrals of some functions related to the degenerate third Painlevé equation, arXiv:1811.05276.
  10. Sloane N.J.A., Sequences A001764, A023745, A029858, A031988, and A014915, The on-line encyclopedia of integer sequences, http://oeis.org.
  11. Suleimanov B.I., Effect of a small dispersion on self-focusing in a spatially one-dimensional case, JETP Lett. 106 (2017), 400-405, arXiv:1706.06849.
  12. Weisstein E.W., Dirichlet divisor problem, Wolfram MathWorld, http://mathworld.wolfram.com/DirichletDivisorProblem.html.
  13. Zhou X., The Riemann-Hilbert problem and inverse scattering, SIAM J. Math. Anal. 20 (1989), 966-986.

Previous article  Next article  Contents of Volume 15 (2019)