Nonexistence of small, odd breathers for a class of nonlinear wave   equations
  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. 
Micha{\l} Kowalczyk, Yvan Martel, and Claudio Mu\~noz
math.AP math-ph math.MP nlin.SI