Eigenvalue-Based Approach to Manipulate and Reconstruct Nonlinear Pulses: Towards Soliton Tomography

Abstract

Soliton content of nonlinear pulses of different physical nature is universally characterized by a discrete set of eigenvalues. In an ideal channel governed by the nonlinear Schrodinger equation, the eigenvalues do not change along the wave field propagation. Perturbations leave predictable fingerprints on the eigenvalue portrait, which was recently used to manipulate optical fiber solitons in [Phys. Rev. Lett. 134, 193804, 2025]. Here, we develop a theoretical framework to manipulate and reconstruct sech-shaped nonlinear wave fields based on soliton eigenvalue response functions and the corresponding inverse problem. We derive analytical expressions to enable nonlinear manipulation of solitons by applying instant, controllable perturbations. Then we present a concept of perturbation sensing with the key feature of nonlinear propagation of the probe signal over an unknown distance, enabling the extraction of information about the perturbation source hidden within nonlinear media or materials. We introduce an integral equation for the inverse problem of reconstructing the unknown shape of the wave field distortions, when the known observational data is a function of deviations in soliton eigenvalues measured at the end of the nonlinear propagation channel. We evaluate different reconstruction regimes and demonstrate a reliable inverse problem solution in presence of noise, paving the way towards soliton tomography.