Experimental emulator of pulse dynamics in fractional nonlinear Schr\"{o}dinger equation

We present a nonlinear optical platform to emulate a nonlinear \textit{L\'{e}vy waveguide} that supports the pulse propagation governed by a generalized fractional nonlinear Schr\"{o}dinger equation (FNLSE). Our approach distinguishes between intra-cavity and extra-cavity regimes, exploring the interplay between the effective fractional group-velocity dispersion (FGVD) and Kerr nonlinearity. In the intra-cavity configuration, we observe stable \textit{fractional solitons} enabled by an engineered combination of the fractional and regular dispersions in the fiber cavity. The soliton pulses exhibit their specific characteristics, \textit{viz.}, "heavy tails" and a "spectral valley" in the temporal and frequency domain, respectively, highlighting the effective nonlocality introduced by FGVD. Further investigation in the extra-cavity regime reveals the generation of spectral valleys with multiple lobes, offering potential applications to the design of high-dimensional data encoding. To elucidate the spectral valleys arising from the interplay of FGVD and nonlinearity, we have developed an innovative "force" model supported by comprehensive numerical analysis. These findings open new avenues for experimental studies of spectral-temporal dynamics in fractional nonlinear systems.