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Sine-gordon EquationThe Sine-Gordon equation is a partial differential equation for a function 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 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 is the angle between the parameter lines. A pseudospherical surface is a surface of negative constant Gaussian curvature . Both PDEs describe solitons. See also Bäcklund transform. The sinh-Gordon equation is given by -
This is the Euler-Lagrange equation of the Lagrangian -
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