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  "body": "\n## Summary\n\nThe Korteweg–De Vries equation, or KdV equation, is the nonlinear [[partial differential equation|GC5BDZ6QBC]] $u_t + u_{xxx} + 6 u u_x = 0$ for a real function $u(x,t)$.\nIt models long waves of small amplitude in shallow water.\nIt has solitary wave solutions that keep their shape, and it is the first equation that was solved by the inverse scattering transform.\nIt has infinitely many conserved quantities and is a central example of a completely integrable system.\n\n## Explanations\n\n### L1 human\n\nImagine a single smooth hump of water moving along a narrow canal without changing its shape.\nTwo effects act on it: the higher parts of the wave move faster, which steepens the front, and dispersion spreads the wave out.\nThe KdV equation contains exactly these two effects, the term $6 u u_x$ and the term $u_{xxx}$.\nWhen they balance, one gets the soliton $u(x,t) = \\tfrac{c}{2} \\mathrm{sech}^2\\!\\big(\\tfrac{\\sqrt{c}}{2}(x - ct)\\big)$ with speed $c \u003e 0$.\nTaller solitons are narrower and move faster.\n",
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