For those with some familiarity with integrable systems, I'll summarize my question as such:
Where can I find literature on ZS-AKNS systems, and their solution via the inverse scattering transform, which goes beyond the special case of the nonlinear Schrödinger equation?
To clarify my meaning, here is some quick review and context. Lately, I've been interested in the $2\times 2$ AKNS integrable system of PDEs (Ablowitz, Kaup, Newell, & Segur 1974) given by $$u_t = u_{xx}+2u^2 v,\hspace{1cm} -v_t = v_{xx}+2v^2 u.\tag{1}$$ (My notation is that used in entry 159, "Nonlinear Schrödinger Equation", of the Encyclopedia of Integrable Systems available online at http://home.itp.ac.ru/~adler/e.html.)
This can be obtained as the compatibility condition $\Phi_{xt}=\Phi_{tx}$ of the Lax pair of evolution equations $\Phi_x = U \Phi$ and $\Phi_t=V\Phi$ where $\Phi=(\phi,\chi)^T$ and $$U=\begin{pmatrix} \lambda & -v \\ u & -\lambda\end{pmatrix},\hspace{1cm} V=-2\lambda U+\begin{pmatrix} -uv & v_x \\ v_x & uv\end{pmatrix},$$ yielding the 'zero-curvature' representation $U_t-V_x+[U,V]=0$ of the system in $(1)$.
The advantage of this formulation is that it is amenable to the inverse scattering transform: For a given pair of potentials $(u,v)$ one 1) determines the scattering data for the eigenvalue problem $\partial_x \Phi =U x$ with $\lambda$ as eigenvalue, 2) allows the scattering data to evolve subject to $\partial_t \Phi=V \Phi$, and 3) reconstructs the potential from the time-evolved scattering data.
However, the literature I've has two limitations. For one, the potentials are assumed to vanish rapidly at infinity; moreover, the literature I've seen tends to focus on $v=\pm u^{*}$ corresponding to the nonlinear Schrodinger equation (either focusing or defocusing). In my case, however I want the two potentials to be independent with $v\to 0$ and $u\to u_{\pm}$ at $x=\pm \infty$.
Hence I'd like to find literature which covers the case of two independent potentials, preferably with discussion of non-vanishing boundary conditions. Are there some useful references?
