Identification of Spines in Nonlinear Fourier Spectra for the Periodic Nonlinear Schrödinger Equation

Identification of Spines in Nonlinear Fourier Spectra for the Periodic Nonlinear Schrödinger Equation: Internship WI5118 - Report

Kitsios, Christos (TU Delft Electrical Engineering, Mathematics and Computer Science)

Wahls, S. (mentor)

Delft University of Technology

Applied Mathematics

2022-03-02

The nonlinear Fourier transform for the focusing periodic nonlinear Schrodinger equation is investigated. This paper is focused on the approximation of the spines in the nonlinear spectrum using results from Floquet theory. Algorithms for the numerical computation of the spines based on the Fourier collocation method are being examined and a new algorithm is presented. The new algorithm developed during the project computes the spines by tracking sign changes of the function ς=(Δ(.)) in the area ℜ<( Δ (.))| < 1, where delta is the Floquet discriminant. The new algorithm is successfully applied to examples where both the modified Fourier collocation method and the method implemented in the FNFT software library fail. In addition, the spine points that are numerically computed by the new algorithm are equally distributed along the curve, while using the other algorithms the computed points are clustered around the periodic eigenvalues. Finally, the algorithm provides information on which spectrum points belong to the same spine. The pseudocode and the MATLAB source code of the algorithm developed are provided.

nonlinear Fourier transform
Nonlinear Schrödinger equation
Nonlinear spectrum

http://resolver.tudelft.nl/uuid:a77a7a6a-5456-4371-b807-f9f64200df7a

Student theses

student report

© 2022 Christos Kitsios