Fourier spectrum and related characteristics of the fundamental bright soliton solution

We derive exact analytical expressions for the spatial Fourier spectrum of the fundamental bright soliton solution for the $(1 + 1)$-dimensional nonlinear Schr\"odinger equation. Similar to a Gaussian profile, the Fourier transform for the hyperbolic secant shape is also shape-preserving. 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 solitons find rich applications in nonlinear fiber optics and optical telecommunication systems.