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  "body": "\n## Summary\n\nThe inverse scattering transform is a method to solve the initial value problem of some nonlinear evolution equations, such as the [[Korteweg–De Vries equation|KTT75QC72Z]].\nThe solution at time $t$ is the potential of a linear spectral problem, such as a [[Schrödinger operator|89AA6XJY5K]] in one variable.\nIts scattering data evolve in time by simple explicit formulas, and the potential is then recovered from the data.\nIt plays for these nonlinear equations the role that the Fourier transform plays for linear equations with constant coefficients.\n\n## Explanations\n\n### L1 human\n\nThe method works in three steps.\nFirst, one turns the initial profile of the wave into spectral data: its bound states and how it reflects incoming waves.\nSecond, one moves these data forward in time, which is easy because each piece evolves on its own.\nThird, one rebuilds the profile from the new data by solving a linear integral equation.\nThe bound states give the [[solitons|63D4AXYCT4]], which is why a localised initial wave splits into a train of solitons and a small spreading tail.\n",
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