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Sine-Gordon equation

From Biocrawler, the free encyclopedia.

The sine-Gordon equation is a partial differential equation in two dimensions. For a function $φ$ of two real variables, x and t, it is

$(\Box + \sin)\phi = \phi_{tt}- \phi_{xx} + \sin\phi = 0.$

The name is a pun on the Klein-Gordon equation, which is

$(\Box + 1)\phi = \phi_{tt}- \phi_{xx} + \phi\ = 0.$

The sine-Gordon equation is the Euler-Lagrange equation of the Lagrangian

$\mathcal{L}_{\mathrm{sine-Gordon}}(\phi) := \frac{1}{2}(\phi_t^2 - \phi_x^2) + \cos\phi.$

If you Taylor-expand the cosine

$\cos(x) = \sum_{n=0}^\infty \frac{(-x^2)^n}{(2n)!}$

and put this into the Lagrangian you get the Klein-Gordon Lagrangian plus some higher order terms

$\mathcal{L}_{\mathrm{sine-Gordon}}(\phi) - 1 = \frac{1}{2}(\phi_t^2 - \phi_x^2) - \frac{\phi^2}{2} + \sum_{n=2}^\infty \frac{(-x^2)^n}{(2n)!} = 2\mathcal{L}_{\mathrm{Klein-Gordon}}(\phi) + \sum_{n=2}^\infty \frac{(-x^2)^n}{(2n)!}$

The sine-Gordon equation has the soliton

$\phi_{\mathrm{soliton}}(x, t) := 4 \arctan \exp(x)\,$

Contents

1 Mainardi-Codazzi equation

2 sinh-Gordon equation

3 External links

4 Bibliography

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Mainardi-Codazzi equation

Another equation is also called the sine-Gordon equation:

$\phi_{uv} = \sin\phi\,$

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 $K = - 1$ .

This partial differential equation has solitons.

sinh-Gordon equation

The sinh-Gordon equation is given by

$\phi_{tt}- \phi_{xx} = -\sinh\phi\,$

This is the Euler-Lagrange equation of the Lagrangian

$\mathcal{L}={1\over 2}(\phi_t^2-\phi_x^2)-\cosh\phi\,$

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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.

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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

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Categories: Partial differential equations | Differential geometry | Equations