On the poles of Picard potentials. (English. Russian original) Zbl 1218.34102

Trans. Mosc. Math. Soc. 2010, 241-250 (2010); translation from Tr. Mosk. Mat. O.-va. 71, 270-282 (2010).
The author studies the existence of a global meromorphic fundamental system of solutions for a system of two differential equations
\[ E_x = \biggl(\begin{matrix} a_{11}z \quad u(x)\\ v(x) \quad a_{22}z\end{matrix}\biggr) E \tag{1} \]
where \(a_{11}\) and \(a_{22}\) are distinct complex numbers. Following Gesztesy and Weikard (1998) a pair \(u(x), v (x)\) of meromorphic functions on \(\mathbb C\) can be defined a Picard potential if, for each \(z\in \mathbb C\), the system (1) has a fundamental system of solutions which is meromorphic on the entire complex plane \(\mathbb C\) of the variable \(x\).
In {Theorem 1} of this paper, the author obtains a necessary condition for an arbitrary pair \(u(x), v (x)\) of meromorphic functions to be a Picard potential, provided that at least one of them has a pole \(x_0 \in \mathbb C\): the function \(u(x), v (x)\) must have, at \(x_0\), a pole of order two with Laurent coefficient of special form. This necessary condition is rather restrictive, but it is still far from being sufficient, which is illustrated by Theorem 2, giving necessary and sufficient condition for the Picard property of monomial functions \(u(x),v (x)\).


34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
Full Text: DOI


[1] A. V. Domrin, The local holomorphic Cauchy problem for soliton equations of parabolic type, Dokl. Akad. Nauk 420 (2008), no. 1, 14 – 17 (Russian); English transl., Dokl. Math. 77 (2008), no. 3, 332 – 335. · Zbl 1163.35300
[2] A. O. Gel\(^{\prime}\)fond, Calculus of finite differences, Hindustan Publishing Corp., Delhi, 1971. Translated from the Russian; International Monographs on Advanced Mathematics and Physics.
[3] Fritz Gesztesy, Karl Unterkofler, and Rudi Weikard, An explicit characterization of Calogero-Moser systems, Trans. Amer. Math. Soc. 358 (2006), no. 2, 603 – 656. · Zbl 1082.37065
[4] Fritz Gesztesy and Rudi Weikard, Elliptic algebro-geometric solutions of the KdV and AKNS hierarchies — an analytic approach, Bull. Amer. Math. Soc. (N.S.) 35 (1998), no. 4, 271 – 317. · Zbl 0909.34073
[5] E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. · Zbl 0063.02971
[6] S. Novikov, S. V. Manakov, L. P. Pitaevskiĭ, and V. E. Zakharov, Theory of solitons, Contemporary Soviet Mathematics, Consultants Bureau [Plenum], New York, 1984. The inverse scattering method; Translated from the Russian. · Zbl 0598.35003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.