Nonlinear wave propagation governed by a fractional derivative

The idea of fractional derivatives has a long history that dates back centuries. Apart from their intriguing mathematical properties, fractional derivatives have been studied widely in physics, for example in quantum mechanics and generally in systems with nonlocal temporal or spatial interactions. However, systematic experiments have been rare because the physical implementation is challenging. Here we report the observation and full characterization of a family of temporal solitons that are governed by a fractional nonlinear wave equation. We demonstrate that these solitons have non-exponential tails, reflecting their nonlocal character, and have a very small time-bandwidth product. The authors experimentally study nonlinear light propagation with tunable dispersion, which mimics the effect of fractional derivatives. The pulses have the unique features that their spectra have a discontinuous derivative and they decay slowly in time.