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  "body": "\n## Summary\n\nA Lax pair is a pair of operators $L(t)$ and $P(t)$ such that an evolution equation is equivalent to $\\frac{dL}{dt} = [P, L] = PL - LP$.\nThen $L(t)$ stays similar to $L(0)$, so its spectrum does not change in time.\nThe eigenvalues of $L$ are thus conserved quantities.\nLax pairs are the starting point of the [[inverse scattering transform|6GY3X4Y8X5]] and a sign that an equation is integrable.\n\n## Explanations\n\n### L1 human\n\nImagine a matrix that changes in time but whose eigenvalues never move.\nA Lax pair writes an equation in exactly this form: the unknown is hidden in the operator $L$, and the equation says $L$ only rotates.\nFor the [[Korteweg–De Vries equation|KTT75QC72Z]], $L$ is the operator $-\\partial_x^2 + u$, and the KdV evolution keeps its eigenvalues fixed.\nThis gives infinitely many conserved quantities and explains why [[solitons|63D4AXYCT4]] keep their speeds after they collide.\n",
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