(Stellenbosch : Stellenbosch University, 2008-12) Olivier, Carel Petrus; Herbst, B. M.; Barashenkov, I. V.; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Applied Mathematics.
The NLS is a universal equation of the class of nonlinear integrable systems. The
aim of this thesis is to study the NLS numerically. More speci cally, an algorithm is
developed to calculate its nonlinear spectrum. The nonlinear spectrum is then used
as a diagnostic for numerical studies of the NLS. The spectrum consists of a discrete
part, further subdivided into the main part, the auxiliary part, and the continuous
spectrum. Two algorithms are developed for calculating the main spectrum. One is
based on Floquet theory, rst implemented by Overman [12]. The other is a direct
calculation of the eigenvalues by Herbst and Weideman [16]. These algorithms
are combined through the marching squares algorithm to calculate the continuous
spectrum. All ideas are illustrated by numerical examples.