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  "body": "\n## Definition\n\nConsider the [[Korteweg–De Vries equation|KTT75QC72Z]] $u_t - 6 u u_x + u_{xxx} = 0$ for $u(x, t)$ real, smooth and decaying fast as $\\abs{x} \\to \\infty$.\nFor each $t$, let $L_t = -\\partial_x^2 + u(\\cdot, t)$ be the one-dimensional [[Schrödinger operator|89AA6XJY5K]] with potential $u(\\cdot, t)$.\nIts *scattering data* $S(t)$ consist of:\n\n1. the finitely many eigenvalues $-\\kappa_1^2, \\dots, -\\kappa_N^2$ of $L_t$, with $\\kappa_n \u003e 0$;\n2. the norming constants $c_n(t) \u003e 0$, where the normalised eigenfunction $\\psi_n$ satisfies $\\psi_n(x) \\sim c_n e^{-\\kappa_n x}$ as $x \\to +\\infty$;\n3. the reflection coefficient $b(k, t)$ for $k \\in \\R$, where a solution of $L_t \\psi = k^2 \\psi$ satisfies $\\psi(x) \\sim e^{-ikx} + b(k, t) e^{ikx}$ as $x \\to +\\infty$ and $\\psi(x) \\sim a(k, t) e^{-ikx}$ as $x \\to -\\infty$.\n\nThe *inverse scattering transform* solves the initial value problem with data $u(x, 0)$ in three steps.\n\n1. Direct scattering: compute $S(0)$ from $u(\\cdot, 0)$.\n2. Time evolution: $\\kappa_n$ stays constant, $c_n(t) = c_n(0) e^{4 \\kappa_n^3 t}$ and $b(k, t) = b(k, 0) e^{8 i k^3 t}$.\n3. Inverse scattering: set $F(x) = \\sum_{n} c_n(t)^2 e^{-\\kappa_n x} + \\frac{1}{2\\pi} \\int_{\\R} b(k, t) e^{ikx} \\, dk$, solve the Gelfand–Levitan–Marchenko equation\n$$K(x, y) + F(x + y) + \\int_x^\\infty K(x, z) F(z + y) \\, dz = 0 \\quad (y \u003e x)$$\nfor $K$, and put $u(x, t) = -2 \\frac{d}{dx} K(x, x)$.\n\n## Remarks\n\nThe time evolution in step 2 comes from the [[Lax pair|G3A4S7CSB3]] of the equation: $L_t$ evolves by conjugation, so its eigenvalues do not move.\n\nExample: take $N = 1$ and $b = 0$, so that $F(x) = c^2 e^{-\\kappa x}$ with $c = c_1(t)$.\nThe kernel $K(x, y) = -\\dfrac{c^2 e^{-\\kappa (x + y)}}{1 + (c^2/2\\kappa) e^{-2\\kappa x}}$ solves the equation.\nWith $w = (c^2/2\\kappa) e^{-2\\kappa x}$ this gives $u = -8\\kappa^2 w/(1 + w)^2 = -2\\kappa^2 \\operatorname{sech}^2(\\kappa(x - x_0))$, where $w = e^{-2\\kappa(x - x_0)}$.\nSince $c^2$ grows like $e^{8\\kappa^3 t}$, the centre is $x_0 = 4\\kappa^2 t + \\text{const}$: this is the one-[[soliton|63D4AXYCT4]] solution moving with speed $4\\kappa^2$.\n\nNon-example: the method needs a Lax pair, and most nonlinear evolution equations have none.\nFor instance, the generalised KdV equation $u_t + u^p u_x + u_{xxx} = 0$ with an integer $p \\ge 3$ is not integrable, and the steps above do not apply to it.\n\nOther equations with a Lax pair, such as the [[nonlinear Schrödinger equation|GEJHRXC176]] and the [[sine-Gordon equation|802QSF9NM3]], have inverse scattering transforms built on other spectral problems.\n\n## Explanations\n\n### L1 human\n\nThe inverse scattering transform turns a hard nonlinear evolution into three linear steps.\nMap the initial wave to spectral data, move the data forward by explicit exponentials, and map back by solving a linear integral equation.\n",
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