A breather is a solution of a nonlinear wave equation or a nonlinear lattice dynamical system which, possibly after passing to a comoving frame, is localized in space and periodic in time. Unlike a traveling soliton, a stationary breather remains localized while its shape or amplitude oscillates, giving the solution its name.
The standard example is a breather solution of the sine-Gordon equation
 |
given, for 
and 
, by
![u(x,t)=4tan^(-1)[(etasin(omegat))/(omegacosh(etax))].](/images/equations/Breather/NumberedEquation2.svg) |
This solution has period 
and approaches 0 as 
.
It can be viewed as a bound soliton-antisoliton
pair, and applying a Lorentz transformation
produces a moving breather (Infeld and Rowlands 2000, pp. 200-201).
Analogous spatially localized, time-periodic solutions in nonlinear lattices are called discrete breathers or intrinsic localized modes. MacKay and Aubry (1994) proved the existence of such solutions for broad classes of weakly coupled oscillator networks. Flach and Willis (1998) give a review.
