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...

Soliton phase shift calculation for the Korteweg–de Vries equation

Several non-linear fluid mechanical processes, such as wave propagation in shallow water, are known to generate solitons: localized waves of translation. Solitons are often hidden in a wave packet at the beginning and only reveal themselves in the far-field. With a special signal processing technique known as the non-linear Fourier transform ...