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  "body": "\n## Summary\n\nThe sine-Gordon equation is the nonlinear wave equation $u_{tt} - u_{xx} + \\sin u = 0$ for a real function $u(x,t)$.\nIts name is a pun on the [[Klein–Gordon equation|AYCCDQWCJF]] $u_{tt} - u_{xx} + u = 0$, which is its linearisation at $u = 0$.\nIt first appeared in the geometry of surfaces of constant negative Gaussian curvature.\nIt is completely integrable and has kink solitons that connect the rest states $0$ and $2\\pi$.\n\n## Explanations\n\n### L1 human\n\nPicture a long row of pendulums hanging from a horizontal wire, with neighbours joined by springs.\nThe angle $u$ of the pendulums along the row obeys the sine-Gordon equation: $\\sin u$ is the pull of gravity and $u_{xx}$ is the pull of the springs.\nA kink is a place where the pendulums make one full turn, and this twist can travel along the row without changing shape.\nThe static kink is $u(x) = 4 \\arctan(e^x)$, which goes from $0$ at $x = -\\infty$ to $2\\pi$ at $x = +\\infty$.\n",
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