Exact solution of the Zakharov–Shabat scattering problem for doubly-truncated multisoliton potentials

Abstract

Recent studies have revealed that multisoliton solutions of the nonlinear Schrödinger equation, as carriers of information, offer a promising solution to the problem of nonlinear signal distortions in fiber optic channels. In any nonlinear Fourier transform based transmission methodology seeking to modulate the discrete spectrum of the multisolitons, choice of an appropriate windowing function is an important design issue on account of the unbounded support of such signals. Here, we consider the rectangle function as the windowing function for the multisolitonic signal and provide a recipe for computing the exact solution of the associated Zakharov–Shabat (ZS) scattering problem for the windowed/doubly-truncated multisoliton potential. The idea consists in expressing the Jost solution of the doubly-truncated multisoliton potential in terms of the Jost solution of the original potential. The proposed method allows us to avoid prohibitive numerical computations normally required in order to accurately quantify the effect of time-domain windowing on the nonlinear Fourier spectrum of the multisolitonic signals. Further, the method devised in this work also applies to general type of signals admissible as ZS scattering potential, and, may prove to be a useful tool in the theoretical analysis of such systems.