Identification of Spines in Nonlinear Fourier Spectra for the Periodic Nonlinear Schrödinger Equation
Internship WI5118 - Report
C. Kitsios (TU Delft - Electrical Engineering, Mathematics and Computer Science)
S. Wahls – Mentor (TU Delft - Team Sander Wahls)
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Abstract
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.