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  "body": "\n## Summary\n\nA Bäcklund transform is a system of first-order partial differential equations that links two functions $u$ and $v$.\nThe system can be solved for $v$ exactly when $u$ solves one [[partial differential equation|GC5BDZ6QBC]].\nThen $v$ solves a second equation.\nWhen the two equations are the same, it is an auto-Bäcklund transform, and it turns known solutions into new ones.\nIt is one of the main tools of the theory of integrable equations and [[solitons|63D4AXYCT4]].\n\n## Explanations\n\n### L1 human\n\nThink of a machine that takes one solution of an equation and outputs another solution.\nThe machine is itself a small system of equations of first order, which is easier to solve than the original equation.\nThe simplest example is the pair of Cauchy–Riemann equations $u_x = v_y$, $u_y = -v_x$.\nIf $u$ is harmonic, these equations give a harmonic conjugate $v$, so they carry one solution of Laplace's equation to another.\nFor the [[sine-Gordon equation|802QSF9NM3]], applying the machine to the zero solution gives a kink, and applying it again gives solutions with two kinks.\n",
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