Reliable numerical algorithms for the Non-linear Fourier Transform of the KdV equation

<b>Research question</b><br/>The topic of this dissertation is the numerical computation of the forward and inverse Non-linear Fourier Transform (NFT) for the Korteweg–de Vries equation (KdV), for sampled signals that decay sufficiently fast on both sides. With NFTs certain non-linear Partial Differential Equations (PDEs) can be solved in a way...

An accurate O(N^2) floating point algorithm for the Crum transform of the KdV equation

We present an algorithm to compute the N-fold Crum transform (also known as the dressing method) for the Korteweg–de Vries equation (KdV) accurately in floating point arithmetic. This transform can be used to generate solutions of the KdV equation, e.g. as a part of the inverse Non-linear Fourier Transform. Crum transform algorithms that...

Improved Fast Inverse Nonlinear Fourier Transform for Multi-solitons: A Discrete Darboux Based Approach

The relation between the input of an ideal optical single-mode fiber and the corresponding fiber output constitutes a nonlinear system that can be described using the nonlinear Schrödinger equation. This nonlinear system has the interesting property that it can be solved analytically using nonlinear Fourier transforms. To utilize this property,...