Abstract
We exhibit some families of Riccati differential equations in the complex domain having elliptic coefficients and study the problem of understanding the cases where there are no multivalued solutions. We give criteria ensuring that all the solutions to these equations are meromorphic functions defined in the whole complex plane, and highlight some cases where all solutions are, furthermore, doubly periodic.
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References
L.V. Ahlfors, Complex analysis, third ed., McGraw-Hill Book Co., New York, 1978.
P. Appel and E. Lacour, Principes de la théorie des fonctions elliptiques et applications, Gauthier-Villars et fils, Paris, 1897.
M. Brunella, Birational geometry of foliations, Publicações Matemáticas do IMPA, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2004.
J. Chazy, Sur les équations différentielles du troisiéme ordre et d’ordre supérieur dont l’intégrale générale a ses points critiques fixes, Acta Math. 34 (1911), no. 1, 317–385.
E. Hille, Ordinary differential equations in the complex domain, Dover Publications, Inc., Mineola, NY, 1997, Reprint of the 1976 original.
E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944.
K. Ishizaki, I. Laine, S. Shimomura, and K. Tohge, Riccati differential equations with elliptic coefficients, Results Math. 38 (2000), no. 1–2, 58–71.
K. Ishizaki, I. Laine, S. Shimomura, and K. Tohge, Riccati differential equations with elliptic coefficients. II, Tohoku Math. J. (2) 55 (2003), no. 1, 99–108.
P. Painlevé, Sur les équations différentielles du second ordre et d’ordre supérieur dont l’intégrale générale est uniforme, Acta Math. 25 (1902), no. 1, 1–85.
S. Shimomura, Meromorphic solutions of a Riccati differential equation with a doubly periodic coefficient, J. Math. Anal. Appl. 304 (2005), no. 2, 644–651.
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This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).
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Guillot, A. Further Riccati Differential Equations with Elliptic Coefficients and Meromorphic Solutions. J Nonlinear Math Phys 25, 497–508 (2018). https://doi.org/10.1080/14029251.2018.1494775
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DOI: https://doi.org/10.1080/14029251.2018.1494775