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  "body": "\n## Definition\n\nConsider an evolution equation for an unknown $u(t)$.\nLet $L(t)$ and $P(t)$ be families of linear operators on a common space, depending on $u(t)$.\nThey form a *Lax pair* for the equation if the equation holds if and only if\n$$\\frac{dL}{dt} = [P, L] = PL - LP.$$\nThe equation $\\frac{dL}{dt} = [P, L]$ is the *Lax equation*.\n\n## Remarks\n\nExample: for $u(x, t)$ smooth, put $L = -\\partial_x^2 + u$ and $P = -4\\partial_x^3 + 6u\\partial_x + 3u_x$, operators acting on functions of $x$.\nA direct computation gives $[P, L] = 6 u u_x - u_{xxx}$, as a multiplication operator, while $\\frac{dL}{dt} = u_t$.\nSo $(L, P)$ is a Lax pair for the [[Korteweg–De Vries equation|KTT75QC72Z]] $u_t - 6 u u_x + u_{xxx} = 0$.\n\nNon-example: if $P = 0$, the Lax equation says that $L$ is constant in $t$.\nWhen $L$ determines $u$, as $L = -\\partial_x^2 + u$ does, a pair $(L, 0)$ is a Lax pair only for the trivial equation $u_t = 0$.\n\nIf $P(t)$ is skew-adjoint and $U(t)$ solves $U' = P U$ with $U(0) = I$, then $L(t) = U(t) L(0) U(t)^{-1}$.\nSo the [[eigenvalues|0BQAA2BK3Q]] of $L(t)$ do not depend on $t$: the flow is *isospectral*.\nFor matrices, $\\tr L(t)^k$ is then constant for each $k$.\n\n## Explanations\n\n### L1 human\n\nA Lax pair rewrites an equation as the statement that an operator $L$ changes only by conjugation.\nConjugation does not change eigenvalues, so the equation has the eigenvalues of $L$ as conserved quantities.\n",
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