Sine-Gordon
Classical field theory
The Sine-Gordon equation corresponds to a Klein-Gordon equation with a source term of the form $\sin(\varphi)$.
\begin{equation} \Box \varphi - g \sin(\varphi) = 0 \end{equation}While highly nonlinear, this can be solved in two dimensions thanks to the inverse scattering transform.
In null coordinates $u,v = (x \pm t)/2$, the Sine-Gordon can be recast as
\begin{equation} \partial_u \partial_v \varphi = g \sin(\varphi(u, v)) \end{equation}A few solutions of the Sine-Gordon equations are known,
Inverse scattering transform
Lax pair :