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  "body": "\n## Definition\n\nLet $E_1$ and $E_2$ be two [[partial differential equations|GC5BDZ6QBC]] for real functions of two variables $(x, y)$.\nA *Bäcklund transform* between $E_1$ and $E_2$ is a system of first-order equations\n$$v_x = F(x, y, u, v, u_x, u_y), \\qquad v_y = G(x, y, u, v, u_x, u_y)$$\nwith these two properties.\n\n1. The compatibility condition $(v_x)_y = (v_y)_x$ of the system, for an unknown $v$, holds exactly when $u$ satisfies $E_1$.\n2. Each pair $(u, v)$ of solutions of the system has $v$ solving $E_2$.\n\nWhen $E_1 = E_2$, it is called an *auto-Bäcklund transform*.\n\n## Remarks\n\nExample: the Cauchy–Riemann system $v_x = -u_y$, $v_y = u_x$ is an auto-Bäcklund transform of Laplace's equation $u_{xx} + u_{yy} = 0$.\nIndeed $(v_x)_y = -u_{yy}$ and $(v_y)_x = u_{xx}$, so the system is compatible exactly when $u_{xx} + u_{yy} = 0$.\nThen $v_{xx} + v_{yy} = -u_{yx} + u_{xy} = 0$ for $C^2$ functions, so $v$ is harmonic too.\n\nExample: for a constant $a \\ne 0$, the system\n$$\\Big(\\frac{v - u}{2}\\Big)_x = a \\sin\\frac{u + v}{2}, \\qquad \\Big(\\frac{u + v}{2}\\Big)_y = \\frac{1}{a} \\sin\\frac{v - u}{2}$$\nis an auto-Bäcklund transform of the [[sine-Gordon equation|802QSF9NM3]] $\\varphi_{xy} = \\sin \\varphi$.\nWith $s = (u + v)/2$ and $d = (v - u)/2$, one gets $d_{xy} = \\cos s \\sin d$ and $s_{xy} = \\cos d \\sin s$, so $u_{xy} = \\sin(s - d) = \\sin u$ and $v_{xy} = \\sin(s + d) = \\sin v$.\n\nNon-example: the system $v_x = u_x$, $v_y = u_y$ is compatible for every $C^2$ function $u$.\nSo it imposes no equation on $u$, and it is not a Bäcklund transform between two equations.\n\n## Explanations\n\n### L1 human\n\nA Bäcklund transform is a pair of first-order equations that ties a new function $v$ to a known function $u$.\nIt can only be solved when $u$ solves a given equation, and then $v$ solves a given equation as well.\n",
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