Consider the Gelfand-Levitan-Marchenko (GLM) integral equation

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

with $F(x)=\sum_{1}^{N} \beta_{n} e^{-c_{n} x}$, where $c_{1}, \ldots, c_{N}$ are positive constants and $\beta_{1}, \ldots, \beta_{N}$ are constants. Consider separable solutions of the form

$$K(x, y)=\sum_{n=1}^{N} K_{n}(x) e^{-c_{n} y}$$

and reduce the GLM equation to a linear system

$$\sum_{m=1}^{N} A_{n m}(x) K_{m}(x)=B_{n}(x)$$

where the matrix $A_{n m}(x)$ and the vector $B_{n}(x)$ should be determined.

How is $K$ related to solutions of the $K d V$ equation?

Set $N=1, c_{1}=c, \beta_{1}=\beta \exp \left(8 c^{3} t\right)$ where $c, \beta$ are constants. Show that the corresponding one soliton solution of the $\mathrm{KdV}$ equation is given by

$$u(x, t)=-\frac{4 \beta_{1} c e^{-2 c x}}{\left(1+\left(\beta_{1} / 2 c\right) e^{-2 c x}\right)^{2}}$$

[You may use any facts about the Inverse Scattering Transform without proof.]