Bright Soliton Solution of the Nonlinear Schrödinger Equation: Fourier Spectrum and Fundamental Characteristics

Natanael Karjanto ()
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Natanael Karjanto: Department of Mathematics, University College, Natural Science Campus, Sungkyunkwan University, Suwon 16419, Republic of Korea

Mathematics, 2022, vol. 10, issue 23, 1-22

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 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.

Keywords: fundamental bright soliton; nonlinear Schrödinger equation; spatial Fourier spectrum; Mittag–Leffler’s expansion theorem; Cauchy’s residue theorem; Parseval’s relation; stretch-bandwidth reciprocity relationship; optical communication systems (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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