Stationary scattering from a nonlinear network

Transmission through a complex network of nonlinear one-dimensional leads is discussed by extending the stationary scattering theory on quantum graphs to the nonlinear regime. We show that the existence of cycles inside the graph leads to a large number of sharp resonances that dominate scattering. The latter resonances are then shown to be extremely sensitive to the nonlinearity and display multistability and hysteresis. This work provides a framework for the study of light propagation in complex optical networks.

A perturbative analysis of dispersion-managed solitons

We study soliton solutions of the path-averaged propagation equation governing the transmission of dispersion-managed (DM) optical pulses in the (practical) limit when residual dispersion and nonlinearity only slightly affect the pulse dynamics over one compensation period. In the case of small dispersion map strengths, the averaged pulse dynamics is governed by a perturbed form of the nonlinear Schrödinger equation; applying a perturbation theory – elsewhere developed – based on inverse scattering theory, we derive an analytic expression for the envelope of the DM soliton. This expression correctly predicts the power enhancement arising from the dispersion management. Theoretical results are verified by direct numerical simulations.

A perturbative analysis of dispersion-managed solitons

We study soliton solutions of the path-averaged propagation equation governing the transmission of dispersion-managed (DM) optical pulses in the (practical) limit when residual dispersion and nonlinearity only slightly affect the pulse dynamics over one compensation period. In the case of small dispersion map strengths, the averaged pulse dynamics is governed by a perturbed form of the nonlinear Schrödinger equation; applying a perturbation theory – elsewhere developed – based on inverse scattering theory, we derive an analytic expression for the envelope of the DM soliton. This expression correctly predicts the power enhancement arising from the dispersion management. Theoretical results are verified by direct numerical simulations.

Appearance of bound states in random potentials with applications to soliton theory

We analyze the stochastic creation of a single bound state (BS) in a random potential with a compact support. We study both the Hermitian Schrödinger equation and non-Hermitian Zakharov-Shabat systems. These problems are of special interest in the inverse scattering method for Korteveg–de-Vries and the nonlinear Schrödinger equations since soliton solutions of these two equations correspond to the BSs of the two aforementioned linear eigenvalue problems. Analytical expressions for the average width of the potential required for the creation of the first BS are given in the approximation of delta-correlated Gaussian potential and additionally different scenarios of eigenvalue creation are discussed for the non-Hermitian case.

Periodic nonlinear Fourier transform for fiber-optic communications, Part I:theory and numerical methods

In this work, we introduce the periodic nonlinear Fourier transform (PNFT) method as an alternative and efficacious tool for compensation of the nonlinear transmission effects in optical fiber links. In the Part I, we introduce the algorithmic platform of the technique, describing in details the direct and inverse PNFT operations, also known as the inverse scattering transform for periodic (in time variable) nonlinear Schrödinger equation (NLSE). We pay a special attention to explaining the potential advantages of the PNFT-based processing over the previously studied nonlinear Fourier transform (NFT) based methods. Further, we elucidate the issue of the numerical PNFT computation: we compare the performance of four known numerical methods applicable for the calculation of nonlinear spectral data (the direct PNFT), in particular, taking the main spectrum (utilized further in Part II for the modulation and transmission) associated with some simple example waveforms as the quality indicator for each method. We show that the Ablowitz-Ladik discretization approach for the direct PNFT provides the best performance in terms of the accuracy and computational time consumption.