Sine-Gordon equation

The sine-Gordon equation is a second-order nonlinear partial differential equation for a function ${\displaystyle \varphi }$ dependent on two variables typically denoted ${\displaystyle x}$ and ${\displaystyle t}$, involving the wave operator and the sine of ${\displaystyle \varphi }$.

It was originally introduced by Edmond Bour (1862) in the course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation for surfaces of constant Gaussian curvature −1 in 3-dimensional space.^[1] The equation was rediscovered by Frenkel and Kontorova (1939) in their study of crystal dislocations known as the Frenkel–Kontorova model.^[2]

This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions,^[3] and is an example of an integrable PDE. Among well-known integrable PDEs, the sine-Gordon equation is the only relativistic system due to its Lorentz invariance.