For the (1+1)-dimensional Klein-Gordon equation called the ${\phi }^4$ model, there is a known asymptotic series formally representing a ‘‘breather’’ (a real-valued solution that is localized in space and periodic in time) in the limit of small amplitude and frequency just below that of spatially uniform infinitesimal oscillations. We show that even though this expansion is valid to all orders, ${\phi }^4$ theory admits no true breathers in this limit. Instead, what appear in many physical contexts are approximate breathers that slowly radiate their energy to x-±∞. We calculate this radiation rate, which lies beyond all orders in the asymptotic expansion.

• Received 1 August 1986

DOI:https://doi.org/10.1103/PhysRevLett.58.747