Sine-Gordon equation

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

Related Topics:
Partial differential equation - Real

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:(Box + sin)phi = phi_{tt}- phi_{xx} + sinphi = 0.

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

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

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

Related Topics:
Euler-Lagrange equation - Lagrangian

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:mathcal{L}_{mathrm{sine-Gordon}}(phi) := rac{1}{2}(phi_t^2 - phi_x^2) + cosphi.

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If you Taylor-expand the cosine

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:cos(x) = sum_{n=0}^infty rac{(-x^2)^n}{(2n)!}

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

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

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::::::: = 2mathcal{L}_{mathrm{Klein-Gordon}}(phi) + sum_{n=2}^infty rac{(-x^2)^n}{(2n)!}

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The sine-Gordon equation has the soliton

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:phi_{mathrm{soliton}}(x, t) := 4 rctan exp(x),

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