The Sine-Gordon equation is a partial differential equation for a function φ of two real variables, x and t, given as follows:
The name is a pun on the Klein-Gordon equation.
.
This is the Euler-Lagrange equation of the Lagrangian
Another equation is also called the Sine-Gordon equation:
where φ is again a function of two real variables u and v.
The last one is better known in the differential geometry of surfaces.
There it is the Mainardi-Codazzi equation , i.e. the integrability condition, of a pseudospherical surface given in (arc-length) asymptotic line parameterization, where φ is the angle between the parameter lines.
A pseudospherical surface is a surface of negative constant Gaussian curvature K = - 1.
Both PDEs describe solitons.
See also Bäcklund transform.
The sinh-Gordon equation is given by
This is the Euler-Lagrange equation of the Lagrangian
External links