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  "body": "\n## Definition\n\nThe *sine-Gordon equation* is the [[partial differential equation|GC5BDZ6QBC]]\n$$u_{tt} - u_{xx} + \\sin u = 0$$\nfor an unknown real function $u(x, t)$ of $x \\in \\R$ and $t \\in \\R$, where subscripts denote [[partial derivatives|09EGC6F8PT]].\n\n## Remarks\n\nExample: the static kink $u(x,t) = 4 \\arctan(e^x)$ is a solution.\nPut $\\theta = \\arctan(e^x)$; then $\\sin 2\\theta = \\mathrm{sech}\\, x$ and $\\cos 2\\theta = -\\tanh x$.\nSo $\\sin u = \\sin 4\\theta = -2\\, \\mathrm{sech}\\, x \\tanh x$.\nAlso $u_x = 2\\, \\mathrm{sech}\\, x$, so $u_{xx} = -2\\, \\mathrm{sech}\\, x \\tanh x = \\sin u$, and $u_{tt} = 0$.\nFor $\\abs{v} \u003c 1$, the moving kink $4 \\arctan\\!\\big(\\exp\\big((x - vt)/\\sqrt{1 - v^2}\\big)\\big)$ is also a solution, by Lorentz invariance.\n\nNon-example: $u(x,t) = x$ is not a solution, because $u_{tt} - u_{xx} + \\sin u = \\sin x$ is not identically $0$.\n\nIn the coordinates $\\xi = (x+t)/2$ and $\\eta = (x-t)/2$ the equation becomes $u_{\\xi\\eta} = \\sin u$.\nThis is the form in which it arises for surfaces of Gaussian curvature $-1$.\n",
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