Gelfand-Levitan-Marchenko equation
= Gelfand-Levitan-Marchenko equation
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{wiki=Marchenko_equation}
For one-dimensional inverse scattering, the Gelfand-Levitan-Marchenko equation
$$
K(x,y)+F(x+y)+\int_x^\infty K(x,z)F(z+y)\,dz=0
$$
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
$$
u(x,t)=-2\frac{\partial}{\partial x}K(x,x;t).
$$