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  "body": "\n## Example\n\nThe [[Korteweg–De Vries equation|KTT75QC72Z]], written here as\n$$u_t + u_{xxx} + 6 u u_x = 0$$\nfor a real function $u(x, t)$ that decays fast as $\\abs{x} \\to \\infty$, is an infinite-dimensional [[integrable system|N0N7CMRCMH]].\n\nIt is Hamiltonian: with\n$$\\mathcal{H}(u) = \\int_{\\R} \\left( \\tfrac{1}{2} u_x^2 - u^3 \\right) dx,$$\nthe variational derivative is $\\delta \\mathcal{H} / \\delta u = -u_{xx} - 3 u^2$, and the equation reads $u_t = \\partial_x \\left( \\delta \\mathcal{H} / \\delta u \\right)$.\n\nIt has infinitely many conserved quantities.\nThe first ones are $\\int u \\, dx$, $\\int u^2 \\, dx$ and $\\int \\left( u^3 - \\tfrac{1}{2} u_x^2 \\right) dx$.\nFor the third one, integration by parts gives $\\frac{d}{dt} \\int u^3 \\, dx = -3 \\int u_x^3 \\, dx$ and $\\frac{d}{dt} \\int \\tfrac{1}{2} u_x^2 \\, dx = -3 \\int u_x^3 \\, dx$, so the difference is constant.\n\nIt has a [[Lax pair|G3A4S7CSB3]]: with the [[Schrödinger operator|W6QS1ME39J]] $L = -\\partial_x^2 - u$, there is a third-order differential operator $B$ such that the equation is equivalent to $L_t = [B, L]$.\nHence the spectrum of $L$ does not change in time.\n\n## Remarks\n\nGardner, Greene, Kruskal and Miura solved the equation in 1967 by the inverse scattering transform.\nThe scattering data of $L$ evolve by explicit linear equations, and $u$ is recovered from them.\nEach eigenvalue of $L$ corresponds to a [[soliton|63D4AXYCT4]], and a general decaying solution splits for large time into solitons and a dispersing tail.\nZakharov and Faddeev showed in 1971 that the scattering data give action-angle variables, in analogy with the [[Liouville–Arnold theorem|PNQCW85HTJ]].\n\n## Explanations\n\n### L1 human\n\nThe KdV equation describes long waves in shallow water.\nIt behaves like a mechanical system with infinitely many degrees of freedom, and it has infinitely many conserved quantities.\nThat is why it can be solved exactly, and why its solitary waves pass through each other and keep their shape.\n",
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