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\let\eps=\varepsilon

Jochen Denzler:
Nonpersistence of breather families for the perturbed sine 
Gordon equation. 
   Comm. Math. Phys., 158(1993), 397-430.

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We show that, up to one exception and as a consequence of
first order perturbation theory only, it is impossible that
a large portion of the well-known family of breather solutions
to the sine Gordon equation could persist under any nontrivial
perturbation of the form
$$
u_{tt} - u_{xx} + \sin u = \eps \Delta(u) + O(\eps^2)
\,,$$
where $\Delta$ is an analytic function in an
{\it arbitrarily small\/}
neighbourhood of $u=0$. Improving known results, we analyze
and overcome the particular difficulties that arise when one
allows the domain of analyticity of $\Delta$ to be small.
The single exception is a one-dimensional linear space
of perturbation functions under which the full family of
breathers does persist up to first order in~$\eps$

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