Sine-Gordon equation

Summary

The sine-Gordon equation is the nonlinear wave equation $u_{tt} - u_{xx} + \sin u = 0$ for a real function $u(x,t)$. Its name is a pun on the Klein–Gordon equation $u_{tt} - u_{xx} + u = 0$, which is its linearisation at $u = 0$. It first appeared in the geometry of surfaces of constant negative Gaussian curvature. It is completely integrable and has kink solitons that connect the rest states $0$ and $2\pi$.