= Solution
Insert the finite exponential expansion into the <Gelfand-Levitan-Marchenko equation>. The integral contribution is
$$
\sum_{m,n}K_m(x)\beta_n e^{-c_ny}\frac{e^{-(c_m+c_n)x}}{c_m+c_n}.
$$
Comparing coefficients of $e^{-c_ny}$ gives
$$
\boxed{A_{nm}(x)=\delta_{nm}+\frac{\beta_n e^{-(c_n+c_m)x}}{c_n+c_m},\qquad B_n(x)=-\beta_n e^{-c_nx}}.
$$
For this coefficient comparison take the $c_n$ distinct; repeated exponents can first be combined. The linear system supplies the coefficients whenever its matrix is invertible.
In the <inverse scattering transform> convention $u_t-6uu_x+u_{xxx}=0$, the <Korteweg-De Vries equation> solution is $u=-2\,dK(x,x)/dx$. The reflectionless scattering data evolve as $\beta_n(t)=\beta_n(0)e^{8c_n^3t}$, with the $c_n$ fixed. For one exponential,
$$
K(x,y)=-\frac{\beta_1e^{-c(x+y)}}{1+(\beta_1/2c)e^{-2cx}}.
$$
Differentiating its diagonal gives
$$
\boxed{u(x,t)=-\frac{4\beta_1c e^{-2cx}}{(1+(\beta_1/2c)e^{-2cx})^2}}.
$$
For $\beta>0$, this is the smooth <soliton>
$$
u=-2c^2\operatorname{sech}^2\!\left(cx-4c^3t-\tfrac12\log(\beta/(2c))\right).
$$
It travels at speed $4c^2$. If $\beta=0$ the solution is zero; if $\beta<0$ the denominator vanishes somewhere, so the formula is not a globally smooth one-soliton. Positivity is the necessary qualification for a regular soliton.
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