Nyquist-Hilbert-nonlinear Schrödinger solitons: A continuous family of fractional nonlinear waves

The applications of fractional derivatives in mathematics and physics are well established. However, experimental observations of waves governed by fractional derivatives have been limited to optical systems under special conditions. Here, we report a theoretical, numerical, and experimental study of a continuous family of fractional nonlinear optical waves that balance Kerr nonlinearity and dispersion of fractional order. We show that within this family, the relationship between pulse width and pulse energy can be freely adjusted and that it connects Nyquist solitons and Hilbert-nonlinear Schrödinger solitons, which were previously investigated separately. These results broaden the fundamental understanding of fractional derivatives and solitons and introduce avenues to engineer novel ultrafast pulses in nonlinear optics and its applications.