Breather ($u(x,t+P)=u(x,t)$)

A breather is a spatially localized nonlinear-wave solution periodic in time. It does not separate into independently traveling lumps while it breathes. A Sine-Gordon breather arises from a complex-conjugate soliton pair. In the quantum sine-Gordon model, the associated neutral kink-antikink bound particles are also called breathers.