  In this note, we show that for a large class of nonlinear wave equations with
odd nonlinearities, any globally defined odd solution which is small in the
energy space decays to $0$ in the local energy norm. In particular, this result
shows nonexistence of small, odd breathers for some classical nonlinear Klein
Gordon equations such as the sine Gordon equation and $\phi^4$ and $\phi^6$
models. It also partially answers a question of Soffer and Weinstein in
\cite[p. 19]{MR1681113} about nonexistence of breathers for the cubic NLKG in
dimension one.
