Direct scattering transform: catch soliton if you can

TL;DR

This paper investigates the numerical errors in the direct scattering transform of soliton wave fields, revealing how domain size affects accuracy and demonstrating reliable analysis of complex wave fields despite inherent limitations.

Contribution

It provides a theoretical analysis of soliton scattering coefficients, uncovering the impact of domain size on numerical errors and the loss of analytical properties in the direct scattering transform.

Findings

Numerical errors depend on the computational domain size L.

An exponential divergence occurs with increasing L in the presence of eigenvalue uncertainties.

Complex wave fields can still be reliably analyzed despite inherent scattering transform limitations.

Abstract

Direct scattering transform of nonlinear wave fields with solitons may lead to anomalous numerical errors of soliton phase and position parameters. With the focusing one-dimensional nonlinear Schr\"odinger equation serving as a model, we investigate this fundamental issue theoretically. Using the dressing method we find the landscape of soliton scattering coefficients in the plane of the complex spectral parameter for multi-soliton wave fields truncated within a finite domain, allowing us to capture the nature of particular numerical errors. They depend on the size of the computational domain $L$ leading to a counterintuitive exponential divergence when increasing $L$ in the presence of a small uncertainty in soliton eigenvalues. In contrast to classical textbooks, we reveal how one of the scattering coefficients loses its analytical properties due to the lack of the wave field compact support in case of $L \to \infty$. Finally, we demonstrate that despite this inherit direct scattering transform feature, the wave fields of arbitrary complexity can be reliably analysed.