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The Sine-Gordon equation is a partial differential equation for a function <math>\phi<math> of two real variables, x and t, given as follows:
- <math>\phi_{tt}- \phi_{xx} = -\sin\phi\,<math>
The name is a pun on the Klein-Gordon equation.
- <math>\phi_{tt}- \phi_{xx} = -\phi\,<math>.
This is the Euler-Lagrange equation of the Lagrangian
- <math>\mathcal{L}={1\over 2}(\phi_t^2-\phi_x^2)+\cos\phi<math>
Another equation is also called the Sine-Gordon equation:
- <math>\phi_{uv} = \sin\phi\,<math>
where <math>\phi<math> 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 <math>\phi<math> is the angle between the parameter lines.
A pseudospherical surface is a surface of negative constant Gaussian curvature <math>K = -1<math>.
Both PDEs describe solitons.
See also Bäcklund transform.
The sinh-Gordon equation is given by
- <math>\phi_{tt}- \phi_{xx} = -\sinh\phi\,<math>
This is the Euler-Lagrange equation of the Lagrangian
- <math>\mathcal{L}={1\over 2}(\phi_t^2-\phi_x^2)-\cosh\phi\,<math>
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