I am trying to work through the Zakharov and Shabat paper on inverse scattering for the nonlinear Schrodinger equation (PDF). I am stuck on section 2.
Problem 1. I need to know how to reconstruct $\Phi$ from its discontinuous definition. I have found the Sokhotskyi formulas which should give $$\Phi(\zeta) = \frac{1}{2 \pi i} \int_{-\infty}^{+ \infty} \frac{\phi (\xi)}{\xi - \zeta} d \xi$$ but this requires $\Phi$ to be analytic in the upper and lower half plane. What I need is some way of extending this to include the poles. The residue of $\Phi$ at each $\zeta = \zeta_k$ is what multiplies each $\frac{1}{\zeta - \zeta_k}$ in the sum. I have no idea why the (1,0) vector is included.
Problem 2. I dont know how the circled equations came about. I dont really know what to say about this, I just dont know how to get them.
Any help you can give me would be much appreciated.
