Sine-Gordon equation
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The sine-Gordon equation is a partial differential equation in two dimensions. For a function <math>\phi<math> of two real variables, x and t, it is
- <math>(\Box + \sin)\phi = \phi_{tt}- \phi_{xx} + \sin\phi = 0.<math>
The name is a pun on the Klein-Gordon equation, which is
- <math>(\Box + 1)\phi = \phi_{tt}- \phi_{xx} + \phi\ = 0.<math>
The sine-Gordon equation is the Euler-Lagrange equation of the Lagrangian
- <math>\mathcal{L}_{\mathrm{sine-Gordon}}(\phi) := \frac{1}{2}(\phi_t^2 - \phi_x^2) + \cos\phi.<math>
If you Taylor-expand the cosine
- <math>\cos(x) = \sum_{n=0}^\infty \frac{(-x^2)^n}{(2n)!}<math>
and put this into the Lagrangian you get the Klein-Gordon Lagrangian plus some higher order terms
- <math>\mathcal{L}_{\mathrm{sine-Gordon}}(\phi) - 1 = \frac{1}{2}(\phi_t^2 - \phi_x^2) - \frac{\phi^2}{2} + \sum_{n=2}^\infty \frac{(-x^2)^n}{(2n)!} = 2\mathcal{L}_{\mathrm{Klein-Gordon}}(\phi) + \sum_{n=2}^\infty \frac{(-x^2)^n}{(2n)!}<math>
The sine-Gordon equation has the soliton
- <math>\phi_{\mathrm{soliton}}(x, t) := 4 \arctan \exp(x)\,<math>
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Mainardi-Codazzi equation
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>.
This partial differential equation has solitons.
See also Bäcklund transform.
sinh-Gordon equation
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>
External links
- Sine-Gordon Equation (http://eqworld.ipmnet.ru/en/solutions/npde/npde2106.pdf) at EqWorld: The World of Mathematical Equations.
- Sinh-Gordon Equation (http://eqworld.ipmnet.ru/en/solutions/npde/npde2105.pdf) at EqWorld: The World of Mathematical Equations.
Bibliography
- A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, 2004.
- R. Rajaraman, Solitons and instantons, North-Holland Personal Library, 1989