Fig. 1: Concept of Hilbert-NLS solitons.
From: Nonlinear wave propagation governed by a fractional derivative
a–e Conventional solitons can form in the presence of quadratic dispersion. a Equivalence of the dispersion operator in the time and frequency domains. b The dispersion (red) and (inverse) group velocity (dashed green) vary smoothly with frequency ω. The dot dashed vertical line marks the soliton central frequency ω0. c At low intensities I, dispersion stretches pulses in time. Solitons can form at high intensities with smooth (d) spectral and (e) temporal profiles that decay exponentially. f Corresponding dispersion operator for Hilbert-NLS solitons. g The dispersion relation Eq. (1) has a discontinuous (disc.) derivative (red) and associated discontinuous (inverse) group velocity (dashed green) as a function of frequency; h At low intensities, the dispersion causes input pulses to split in two. At high intensities, solitons form, which have (i) a spectrum with discontinuous derivative, and (k) non-exponential decay in time. The units are arbitrary, and the intensity profiles in (d, e, i) and (k) are on logarithmic scales.
