Processing of optical signals by "surgical" methods for the Gelfand-Levitan-Marchenko equation

TL;DR

The paper introduces a novel 'surgical' method for solving the Gelfand-Levitan-Marchenko equation using a block Toeplitz approach, enabling stable solutions at arbitrary points for complex soliton signals.

Contribution

It presents a new computational technique that improves stability and flexibility in solving the GLME for multi-soliton signals.

Findings

Method successfully computes solutions for two solitons.

Method applicable to complex eight-soliton configurations.

Avoids numerical instability in soliton calculations.

Abstract

We propose a new method for solving the Gelfand-Levitan-Marchenko equation (GLME) based on the block version of the Toeplitz Inner-Bordering (TIB) with an arbitrary point to start the calculation. This makes it possible to find solutions of the GLME at an arbitrary point with a cutoff of the matrix coefficient, which allows to avoid the occurrence of numerical instability and to perform calculations for soliton solutions spaced apart in the time domain. Using an example of two solitons, we demonstrate our method and its range of applicability. An example of eight solitons shows how the method can be applied to a more complex signal configuration.