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Journal of Mathematical Physics / Volume 53 / Issue 2 / ARTICLES / Methods of Mathematical Physics

J. Math. Phys. 53, 023505 (2012); http://dx.doi.org/10.1063/1.3682692 (5 pages)

A note on the first integrals of the ABC system

Jaume Llibre1 and Clàudia Valls2

1Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
2Departamento de Matemática, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais 1049-001, Lisboa, Portugal

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(Received 20 July 2011; accepted 17 January 2012; published online 8 February 2012)

Without loss of generality the ABC systems reduce to two cases: either A = 0 and B, C ⩾ 0, or A = 1 and 0 < B, C ⩽ 1. In the first case it is known that the ABC system is completely integrable, here we provide its explicit first integrals. In the second case Ziglin [“Dichotomy of the separatrices and the nonexistence of first integrals in systems of differential equations of Hamiltonian type with two degrees of freedom,” Izv. Akad. Nauk SSSR, Ser. Mat. 51, 1088 (1987)] proved that the ABC system with 0 < B < 1 and C > 0 sufficiently small has no real meromorphic first integrals. We improve Ziglin's result showing that there are no C1 first integrals under convenient assumptions.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. PROOF OF PROPOSITION 1
  3. PROOF OF THEOREM 1
  4. PROOF OF THEOREM 3

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KEYWORDS and PACS

Keywords

integral equations, partial differential equations

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3682692

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
    Friedlander, S. and Vishik, M., “Instability criteria for the flow of an inviscid incompressible fluid,” Phys. Rev. Lett. 66, 2204–2206 (1991).

    Friedlander, S. and Vishik, M., “Instability criteria for steady flows of a perfect fluid,” Chaos 2, 455–460 (1992)CHAOEH000002000003000455000001.

    Ziglin, S. L., “On the absence of a real-analytic first integral for ABC flow when A = B,” Chaos 8, 272–273 (1998)CHAOEH000008000001000272000001.



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