The sine-Gordon equation

\begin{displaymath}u_{xt} = \sin (u) \end{displaymath}

describes the surfaces of constant negative curvature.

These pictures are the surfaces that correspond to soliton and breather solutions of this equation.

In my paper "Sine-Gordon equation and representations of $\hat sl_2$", I show that the hidden symmetries of the sine-Gordon equation are given by the Kac-Moody algebra $\hat sl_2$ and obtain the soliton solutions via the Kac-Moody group action on the space of differential operators.

The images were generated with Richard Palais' 3DFilmstrip program by Robert Milson (Dalhousie University).