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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Elliptic solitons, Fuchsian equations, and algorithms


Author: Yu. V. Brezhnev
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 24 (2012), nomer 4.
Journal: St. Petersburg Math. J. 24 (2013), 555-574
MSC (2010): Primary 35C08; Secondary 35J10
Published electronically: May 24, 2013
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Abstract: It is shown how the elliptic finite-gap potentials of the Schrödinger equation give rise to a family of solvable linear differential equations of the Fuchs class on the plane and on the torus: the latter case cannot be integrated via realizations of the Zinger-Kovacic type algorithms known in the Picard-Vessiot theory. For the arising Fuchsian equations, monodromy groups and their representations are constructed, the differential Galois group is described, together with a (recursive) method for calculation of the objects involved therein.


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Additional Information

Yu. V. Brezhnev
Affiliation: Tomsk State University, Lenin ave. 36, Tomsk 634050, Russia
Email: brezhnev@mail.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-2013-01253-2
PII: S 1061-0022(2013)01253-2
Keywords: Elliptic solitons, Fuchsian equations, monodromy groups, integration methods, Kovacic algorithm
Received by editor(s): April 5, 2011
Published electronically: May 24, 2013
Additional Notes: Supported by the Federal Program “Scientific and pedogogical personnel of innovative Russia”, grant no. 14.B37.21.0911.
Article copyright: © Copyright 2013 American Mathematical Society