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Keywords = stretch-bandwidth reciprocity relationship

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22 pages, 523 KB  
Review
Bright Soliton Solution of the Nonlinear Schrödinger Equation: Fourier Spectrum and Fundamental Characteristics
by Natanael Karjanto
Mathematics 2022, 10(23), 4559; https://doi.org/10.3390/math10234559 - 1 Dec 2022
Cited by 5 | Viewed by 3306
Abstract
We derive exact analytical expressions for the spatial Fourier spectrum of the fundamental bright soliton solution for the 1+1-dimensional nonlinear Schrödinger equation. Similar to a Gaussian profile, the Fourier transform for the hyperbolic secant shape is also shape-preserving. Interestingly, this [...] Read more.
We derive exact analytical expressions for the spatial Fourier spectrum of the fundamental bright soliton solution for the 1+1-dimensional nonlinear Schrödinger equation. Similar to a Gaussian profile, the Fourier transform for the hyperbolic secant shape is also shape-preserving. Interestingly, this associated hyperbolic secant Fourier spectrum can be represented by a convergent infinite series, which can be achieved using Mittag–Leffler’s expansion theorem. Conversely, given the expression of the series of the spectrum, we recover its closed form by employing Cauchy’s residue theorem for summation. We further confirm that the fundamental soliton indeed satisfies essential characteristics such as Parseval’s relation and the stretch-bandwidth reciprocity relationship. The fundamental bright soliton finds rich applications in nonlinear fiber optics and optical telecommunication systems. Full article
(This article belongs to the Special Issue Novel Mathematical Methods in Signal Processing and Its Applications)
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