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Gelfand-Levitan-Marchenko equation

For one-dimensional inverse scattering, the Gelfand-Levitan-Marchenko equation

K (x, y) + F (x + y) + \int_{x}^{\infty} K (x, z) F (z + y) d z = 0

recovers the transformation kernel

K

from scattering data encoded by

F

. In the convention used for the Korteweg-De Vries equation above, the potential is

u (x, t) = - 2 \frac{\partial}{\partial x} K (x, x; t) .

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