= Solution
Put
$$
c=\beta _0e^{8\chi^3t},
\qquad F(s)=ce^{-\chi s},
$$
and seek the <rank-one Gelfand-Levitan-Marchenko kernel> in the separable form
$$
K(x,y)=-B(x)e^{-\chi y}.
$$
Substitution into the <Gelfand-Levitan-Marchenko equation> gives
$$
-B(x)+ce^{-\chi x}
-B(x)c\int_x^\infty e^{-2\chi z}\,dz=0,
$$
so
$$
B(x)=\frac{ce^{-\chi x}}
{1+ce^{-2\chi x}/(2\chi)}
$$
and hence
$$
K(x,x)=-\frac{ce^{-2\chi x}}
{1+ce^{-2\chi x}/(2\chi)}.
$$
Write
$$
q=\frac{ce^{-2\chi x}}{2\chi}
=\frac{\beta _0}{2\chi}e^{8\chi^3t-2\chi x}.
$$
The reconstruction formula for this <inverse scattering transform> convention yields
$$
u(x,t)=-2\frac{\partial}{\partial x}K(x,x)
=-\frac{8\chi^2q}{(1+q)^2}.
$$
If $q=e^{-2s}$, then $4q/(1+q)^2=\operatorname{sech}^2s$. Taking
$$
s=\chi(x-4\chi^2t-\phi)
$$
therefore produces the required one-<soliton> of the <Korteweg-De Vries equation>, with
$$
\boxed{\phi=\frac{1}{2\chi}\log\!\left(\frac{\beta _0}{2\chi}\right)}.
$$
Here the usual bound-state norming constant has $\beta _0>0$ and $\chi>0$, so $\phi$ is real.
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