For one-dimensional inverse scattering, the Gelfand-Levitan-Marchenko equationrecovers the transformation kernel from scattering data encoded by . In the convention used for the Korteweg-De Vries equation above, the potential is
If with , the Gelfand-Levitan-Marchenko equation has a separable kernel. Writing reduces the integral equation to one algebraic equation for and reconstructs a single soliton.
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The Marchenko equations are a set of integral equations used in the mathematical and physical analysis of wave propagation, particularly in the field of scattering theory and inverse problems. They are named after the Russian mathematician Vladimir Marchenko. The Marchenko equations are typically used to reconstruct the potential in one-dimensional quantum mechanical systems from scattering data.