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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A hypergeometric function approach
to the persistence problem
of single sine-Gordon breathers


Author: Jochen Denzler
Journal: Trans. Amer. Math. Soc. 349 (1997), 4053-4083
MSC (1991): Primary 35Q53; Secondary 33C05, 35B10, 44A10
MathSciNet review: 1422601
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Abstract: It is shown that for an interesting class of perturbation functions, at most one of the continuum of sine-Gordon breathers can persist for the perturbed equation. This question is much more subtle than the question of persistence of large portions of the family, because analytic continuation arguments in the amplitude parameter are no longer available. Instead, an asymptotic analysis of the obstructions to persistence for large Fourier orders is made, and it is connected to the asymptotic behaviour of the Taylor coefficients of the perturbation function by means of an inverse Laplace transform and an integral transform whose kernel involves hypergeometric functions in a way that is degenerate in that asymptotic analysis involves a splitting monkey saddle. Only first order perturbation theory enters into the argument. The reasoning can in principle be carried over to other perturbation functions than the ones considered here.


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Additional Information

Jochen Denzler
Affiliation: Mathematisches Institut, Ludwig–Maximilians–Universität, Theresienstraße 39, D–80333 München, Germany; Lefschetz Center of Dynamical Systems, Brown University, Providence, RI 02906
Email: denzler@rz.mathematik.uni-muenchen.de

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01951-X
PII: S 0002-9947(97)01951-X
Keywords: Sine-Gordon equation, breather, Laplace transform, hypergeometric function, saddle point analysis
Received by editor(s): October 18, 1995
Received by editor(s) in revised form: March 25, 1996
Article copyright: © Copyright 1997 American Mathematical Society