# Literature on ZS-AKNS systems with independent potentials

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?

• Just a clarification: the Lax Pair and the AKNS are equivalent, but distinct approaches. The AKNS scheme has the advantage of it being somewhat easier to get the operators you need. – Adrian Keister Feb 20 '18 at 20:30

Beals and Coifman provided a general analysis of the IST for the $N\times N$ version of the AKNS problem (see Comm. Pure Appl. Math. 37 (1984) 39-90 and their subsequent papers).

Their analysis appears to meet your requirement for the independence of the $u$ and $v$ potentials. However, your requirement for non-vanishing boundary conditions is not met as far as I can see.

http://www.sciencedirect.com.sci-hub.cc/science/article/pii/037596019090576A

If you can't see it here is the title

"The ZS-AKNS inverse scattering transform with poles of multiple orders"

the author

"Zhuhan Jiang"

and the doi

10.1016/0375-9601(90)90576-A