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\let\eps=\varepsilon



Jochen Denzler:
Second order nonpersistence of the sine Gordon breather under
an exceptional perturbation.
   Annales de l'Institut Henri Poincar\'e, Analyse non lin\'eaire,
   12(1995), 201-239

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We consider the only non-trvial perturbation of the sine Gordon
equation of the type 
$$
u_{tt} - u_{xx} + \sin u = \eps \Delta(u) + O(\eps^2)
$$
under which persistence of the unperturbed breather family
cannot be ruled out by first order perturbation theory.
We show that in this case, nonpersistence can be ruled out
by second order perturbation theory. A resonant interaction of
the second order perturbation function with the first order
perturbation of the breathers is responsible for this phenomenon.
Number theoretic techniques make the final analysis manageable. 

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