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Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

 
 

 

On the poles of Picard potentials


Author: A. V. Komlov
Translated by: Alex Martsinkovsky
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 71 (2010).
Journal: Trans. Moscow Math. Soc. 2010, 241-250
MSC (2010): Primary 34M45; Secondary 35Q51, 35Q53
DOI: https://doi.org/10.1090/S0077-1554-2010-00182-3
Published electronically: December 21, 2010
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the existence of a global meromorphic fundamental system of solutions for a system of two differential equations $ E_{x} = (az +q(x))E$, where $ a$ is a constant diagonal matrix, and $ q(x)$ is an off-diagonal meromorphic function, for each $ z \in \mathbb{C}$. Following Gesztesy and Weikard (1998), who investigated this property of functions $ q(x)$ and its connection to finite-gap solutions of soliton equations, we call such $ q(x)$ Picard potentials. We obtain conditions for the Picard property of various potentials $ q(x)$.


References [Enhancements On Off] (What's this?)

  • 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) MR 2462096 (2009g:35251)
  • 2. A. O. Gel'fond, Calculus of finite differences, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1959; English transl., International Monographs on Advanced Mathematics and Physics. Hindustan Publishing Corp., Delhi, 1971. MR 0342890 (49:7634)
  • 3. F. Gesztesy, K. Unterkofler, and R. Weikard, An explicit characterization of Calogero-Moser systems, Trans. Amer. Math. Soc. 358 (2006), no. 2, 603-656. MR 2177033 (2006h:35229)
  • 4. F. Gesztesy and R. 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. MR 1638298 (99i:58075)
  • 5. E. L. Ince, Ordinary differential equations, Dover Publications, New York, 1944. MR 0010757 (6:65f)
  • 6. S. V. Manakov, S. P. Novikov, L. P. Pitaevskiĭ, and V. E. Zakharov, Theory of solitons. The inverse scattering method. Nauka, Moscow, 1980; English transl., Contemporary Soviet Mathematics, Consultants Bureau [Plenum], New York, 1984. MR 779467 (86k:35142)

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Additional Information

A. V. Komlov
Affiliation: M. V. Lomonosov Moscow State University
Email: komlov@hotbox.ru

DOI: https://doi.org/10.1090/S0077-1554-2010-00182-3
Published electronically: December 21, 2010
Additional Notes: Supported by the RFFI Grant 08–01–00014 and by the Support Program for Leading Scientific Schools Grant NSh-3877.2008.1
Article copyright: © Copyright 2010 American Mathematical Society

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