Direct scattering transform: catch soliton if you can

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 Schrödinger 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.

arXiv comment6 pages, 3 figures

References in corpus (20)

• Information Transmission using the Nonlinear Fourier Transform, Part I: Mathematical Tools
• Integrable turbulence and formation of rogue waves
• Bound state soliton gas dynamics underlying the noise-induced modulational instability
• The role of multiple soliton and breather interactions in generation of rogue waves: the mKdV framework
• Strongly interacting soliton gas and formation of rogue waves
• Statistical properties of nonlinear stage of modulation instability in fiber optics
• Integrable turbulence generated from modulational instability of cnoidal waves
• Nonlinear spectral analysis of Peregrine solitons observed in optics and in hydrodynamic experiments
• Nonlinear random optical waves: integrable turbulence, rogue waves and intermittency
• Formation of rogue waves from the locally perturbed condensate
• New approaches to coding information using inverse scattering transform
• Auto-modulation versus breathers in the nonlinear stage of modulational instability
• Exponential Fourth Order Schemes for Direct Zakharov-Shabat problem
• Statistics of the Nonlinear Discrete Spectrum of a Noisy Pulse
• Direct nonlinear Fourier transform algorithms for the computation of solitonic spectra in focusing nonlinear Schrödinger equation
• Nonlinear propagation of an optical speckle field
• Higher Order Convergent Fast Nonlinear Fourier Transform
• Soliton-based discriminator of non-coherent optical pulses
• Exact Solution of the Zakharov-Shabat Scattering Problem for Doubly-Truncated Multi-Soliton Potentials
• Novel Numerical Algorithm with Fourth-Order Accuracy for the Direct Zakharov-Shabat Problem

Cited by in corpus (5)

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• Solitons in a box-shaped wavefield with noise: perturbation theory and statistics
• Bound-state soliton gas as a limit of adiabatically growing integrable turbulence
• Properties of synchronous collisions of solitons in the Korteweg - de Vries equation
• Successive Eigenvalue Removal for Multi-Soliton Spectral Amplitude Estimation