Scattering data and IST

Question

I'm looking into the IST (inverse scattering transform), in particular, its application to the kdv eq.

I've picked a particular i.c. (though I don't think it's relevant to my question) for the schrodinger scattering problem $ \phi_{xx}+(\lambda-u(x,0)) \cdot \phi =0 $

and for it I'm looking for the initial scattering data which is necessary for ist.

In my computation I derived separately the solution do to the continuous spectrum and that do to the discrete spectrum, but I think I have stumbled upon a fairly trivial connection between the two and I'm looking for conformation.

is it true that $b_n(\kappa_n)=b(k=i \cdot \kappa_n)$?

where the continuous solution is-

$\phi(x\to\infty)\approx a*e^{-ikx}+b*e^{ikx} $

$\phi_n(x\to-\infty)\approx e^{-ikx} $

and the discrete-

$\phi_n(x\to\infty)\approx b_n*e^{-\kappa_nx} $

$\phi_n(x\to-\infty)\approx e^{\kappa_nx} $

Thanks, E

Answer 1

After due consideration (and help from the course lecturer) it has been decided that the statement is false, this wishful thinking disregards branching points of b. :(