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>

Contents

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Origin of the equation and name

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(\phi) = \sum_{n=0}^\infty \frac{(-\phi ^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{(-\phi^2)^n}{(2n)!}</math>
<math> = 2\mathcal{L}_{\mathrm{Klein-Gordon}}(\phi) + \sum_{n=2}^\infty \frac{(-\phi^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.

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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>
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External links

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Bibliography

  • A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2004. ISBN 1-58488-355-3
  • R. Rajaraman, Solitons and instantons, North-Holland Personal Library, 1989
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