Wave turbulence theory provides a unifying framework for analyzing the nonlinear interactions of dispersive waves across diverse physical regimes. These interactions manifest over an extensive range of scales and media, encompassing phenomena such as gravitational waves in astrophysical contexts, surface waves in fluid dynamics, coherent structures in quantum systems, and even pattern formation in everyday fluid flows.
Despite their widespread occurrence, the mathematical analysis of such systems remains exceptionally challenging due to the interplay between nonlinearity, dispersion, and randomness. These difficulties have catalyzed the development of sophisticated analytical and probabilistic tools, drawing from partial differential equations, statistical mechanics, and dynamical systems theory.
In this lecture, I will present a short survey of recent advances in the rigorous treatment of wave turbulence, emphasizing key techniques, foundational results, and persistent open problems. Some attention will be given to the derivation and analysis of kinetic equations, scaling limits, and the role of resonant interactions in the long-time dynamics of weakly nonlinear wave systems.
