Rogue Waves in Integrable Turbulence: Semi-Classical Theory and Fast Measurements

Integrable turbulence arises while noisy initial data are launched into a system described by an integrable wave equation. Over recent years, the study of the complex dynamics and non-Gaussian statistics in integrable turbulence has become a growing and fundamental field of research. In this chapter, we show how recent theoretical results obtained in the semi-classical (zero-dispersion) limit of the focusing one dimensional nonlinear Schrödinger equation (NLSE) can be related to the phenomenon of integrable turbulence. One of the outcomes of the semi-classical NLSE analysis is an asymptotic description of the evolution in the framework of finite-gap theory, also known as periodic inverse scattering transform (IST), and we show how this periodic IST can be used to characterize integrable turbulence. Finally, we review optical fiber experiments devoted to the study of (near) integrable turbulence. In particular, we show how fast measurement techniques (time lens, time microscope, optical sampling) now allow the detailed study of the complex and fast dynamics arising from the propagation of partially coherent waves in optical fibers. These experiments provide physical confirmation of some fundamental predictions of the semi-classical theory.