Numerical Algorithms for the Direct Spectral Transform with Applications to Nonlinear Schrodinger Type Systems

We implement two different algorithms for computing numerically the direct Zakharov–Shabat eigenvalue problem on the infinite line. The first algorithm replaces the potential in the eigenvalue problem by a piecewise-constant approximation, which allows one to solve analytically the corresponding ordinary differential equation. The resulting algorithm is of second order in the step size. The second algorithm uses the fourth-order Runge–Kutta method. We test and compare the performance of these two algorithms on three exactly solvable potentials. We find that even though the Runge–Kutta method is of higher order, this extra accuracy can be lost because of the additional dependence of its numerical error on the eigenvalue. This limits the usefulness of the Runge–Kutta algorithm to a region inside the unit circle around the origin in the complex plane of the eigenvalues. For the computation of the continuous spectrum density, this limitation is particularly severe, as revealed by the spectral decomposition of the L2-norm of a solution to the nonlinear Schrödinger equation. We show that no such limitations exist for the piecewise-constant algorithm. In particular, this scheme converges uniformly for both continuous and discrete spectrum components. c © 1998 Academic Press