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  "body": "\n## Definition\n\nThe *Korteweg–De Vries equation* is the [[partial differential equation|GC5BDZ6QBC]]\n$$u_t + u_{xxx} + 6 u u_x = 0$$\nfor an unknown real function $u(x, t)$ of $x \\in \\R$ and $t \\in \\R$.\nHere subscripts denote [[partial derivatives|09EGC6F8PT]], for example $u_{xxx} = \\partial_x^3 u$.\n\n## Remarks\n\nExample: for $c \u003e 0$, the function $u(x,t) = f(x - ct)$ with $f(\\xi) = \\tfrac{c}{2} \\mathrm{sech}^2\\!\\big(\\tfrac{\\sqrt{c}}{2}\\xi\\big)$ is a solution.\nSubstituting $u = f(x - ct)$ gives $-c f' + f''' + 6 f f' = 0$, and after one integration $f'' = c f - 3 f^2$.\nWith $b = \\sqrt{c}/2$ and $s = \\mathrm{sech}(b\\xi)$, one computes $f'' = \\tfrac{c}{2} b^2 (4 s^2 - 6 s^4) = \\tfrac{c^2}{2} s^2 - \\tfrac{3c^2}{4} s^4$, which equals $c f - 3 f^2$.\n\nNon-example: $u(x,t) = x$ is not a solution, because $u_t + u_{xxx} + 6 u u_x = 6x \\ne 0$.\n\nOther normalisations are common, such as $u_t + u_{xxx} - 6 u u_x = 0$, obtained by replacing $u$ with $-u$.\nAny equation $u_t + \\alpha u_{xxx} + \\beta u u_x = 0$ with $\\alpha, \\beta \\ne 0$ reduces to this form by scaling $x$, $t$ and $u$.\n",
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