Inverse scattering transform and GLM (Gel'fand-Levitan-Marchenko) equation

Question

Can anyone please explain how to derive the GLM equation and why one can recover the potential using just the scattering data?

Answer 1

You need to go to Ablowitz and Segur, Solitons and the Inverse Scattering Transform, p. 20ff, and also p. 48. The idea is that you solve the Gel'fand-Levitan-Marchenko integral equation for its kernel $K(x,y;t)$, independent of the eigenvalue: $$K(x,y;t)\mp F^*(x+y;t)\pm\int_x^{\infty}\int_x^{\infty}K(x,z;t)\,F(y+s;t)\,F^*(s+z;t)\,ds\,dz=0,$$ where you compute $F$ as the following: $$F(x;t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{b}{a}(\xi,t)\,e^{i\xi x}\,d\xi-i\sum_{j=1}^NC_j(t)\,e^{i\xi_j x}.$$ Note that the scattering data shows up in the definition of $F:\;b/a$ is the reflection coefficient, the $C_j(t)$ are the norming constants, and the $\xi_j$ are the eigenvalues. Once you have the kernel, the final solution to the overall problem is $q(x,t)=-2K(x,x;t)$.