Sine-Gordon equation

The sine-Gordon equation is a second-order nonlinear partial differential equation for a function

φ

{\displaystyle \varphi }

dependent on two variables typically denoted

x

{\displaystyle x}

and

t

{\displaystyle t}

, involving the wave operator and the sine of

φ

{\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. The equation was rediscovered by Yakov Frenkel and Tatyana Kontorova (1939) in their study of crystal dislocations known as the Frenkel–Kontorova model.

This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions, 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.

Realizations of the sine-Gordon equation

Differential geometry

This is the first derivation of the equation, by Bour (1862).

There are two equivalent forms of the sine-Gordon equation. In the (real) space-time coordinates, denoted

(

x

,

t

)

{\displaystyle (x,t)}

, the equation reads:

φ

t

t

−

φ

x

x

+

sin

⁡

φ

=

0

,

{\displaystyle \varphi _{tt}-\varphi _{xx}+\sin \varphi =0,}

where partial derivatives are denoted by subscripts. Passing to the light-cone coordinates (u, v), akin to asymptotic coordinates where

u

=

x

+

t

2

,

v

=

x

−

t

2

,

{\displaystyle u={\frac {x+t}{2}},\quad v={\frac {x-t}{2}},}

the equation takes the form

φ

u

v

=

sin

⁡

φ

.

{\displaystyle \varphi _{uv}=\sin \varphi .}

This is the original form of the sine-Gordon equation, as it was considered in the 19th century in the course of investigation of surfaces of constant Gaussian curvature K = −1, also called pseudospherical surfaces.

Consider an arbitrary pseudospherical surface.