= Solution
Each discrete term satisfies
$$
\partial_t(c_n^2e^{-\kappa_nx})=8\kappa_n^3c_n^2e^{-\kappa_nx}
=-8\partial_x^3(c_n^2e^{-\kappa_nx}),
$$
and the same identity follows for the Fourier integral from $R_t=8ik^3R$. Hence
$$
F_t+8F_{xxx}=0.
$$
Apply $\partial_x^2-\partial_y^2-u(x,t)$ to the <Gelfand-Levitan-Marchenko equation>. Differentiating under the integral and integrating the $z$ derivatives by parts, the boundary terms combine to $u=-2\partial_xK(x,x,t)$. The terms containing $F$ cancel, leaving
$$
\boxed{G(x,y,t)+\int_x^\infty G(x,z,t)F(z+y,t)\,dz=0}.
$$
Uniqueness of the Marchenko equation then also gives $G=0$.
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