Finite difference methods for an AKNS eigenproblem (Q676567)

The authors consider the numerical solution of the eigenvalue problem (EVP) \[ v'(x)+ i\zeta v(x) =q(x)w(x), \quad w'(x)-i \zeta w(x)= -\overline q(x)v(x). \] Here \(q(x)\) is a given potential of period \(2\pi\), \(v(x)\) and \(w(x)\) are the eigenfunctions, and \(\zeta\) is the eigenvalue, all of which may be complex. The boundary conditions are \(v(x+2\pi) =v(x)\), \(w(x+2 \pi) =w(x)\) (periodic solutions), and \(v(x+2 \pi)= -v(x)\), \(w(x+2 \pi)= -w(x)\) (antiperiodic solutions). This EVP is a special case of the AKNS problem introduced by \textit{M. J. Ablowitz}, \textit{D. J. Kaup}, \textit{A. C. Newell} and \textit{H. Segur} [Studies Appl. Math. 53, 249-315 (1974; Zbl 0408.35068)] in the context of the inverse scattering method, a procedure for generating explicit solutions to some nonlinear soliton equations. The present authors are interested in numerical procedures that yield approximations to a large part of the spectrum. They apply finite difference schemes to the test problem with the single Fourier mode potential \(q(x)= A \exp (i\mu x)\), \(\mu=0\), \(\pm 1\), \(\pm 2, \dots\). In this case the exact eigensolutions are explicitly known. They compare three finite difference schemes with respect to accuracy and effectiveness, and they present an improved version of the Ablowitz-Ladik scheme.