Numerical algorithms for the direct spectral transform with applications to nonlinear Schrödinger type systems
In the context of the nonlinear Schrödinger equation which describes propagation of light pulses in an optical fiber, the direct Zakharov-Shabat spectral problem plays an important role. It reads \[ \Phi_1'= q\Phi_2- i\zeta\Phi_1,\quad \Phi_2'= -q^*\Phi_1+ i\zeta\Phi_2 \] together with some asymptotic conditions. Here, \(\Phi_1\) and \(\Phi_2\) are scalar wave functions, \(q\) is a potential, and \(\zeta\) an eigenvalue parameter. Two different algorithms were implemented for solving numerically the eigenvalue problem on the infinite line. The first algorithm replaces the potential by a piecewise constant approximation, which allows one to solve analytically the corresponding ordinary differential equation. The second algorithm uses the fourth-order Runge-Kutta method. The authors find that even though the Runge-Kutta method is of higher-order, this accuracy can be lost because of the additional dependence of its numerical error on the eigenvalue, whereas the first scheme converges uniformly for both continuous and discrete spectrum components.
scientific article; zbMATH DE number 1282745