Breathers solution of the sine-Gordon equation

Breathers are spatially localised time-periodic.

$$u(x,t) = 4 \arctan{\frac{\beta \sin{\omega t}}{\omega \cosh{\beta(x-x_0)}}},$$

with $\omega^2 + \beta^2 = 1$, discovered by Ablowitz, Kaup, Newell and Segur [#!AKNS!#] and called ``breathers'' by them because of their behaviour when $\omega$ is small.

 

Figure 3: Breather

 

Figure 4: Moving Breather