Integrability and Hydrodynamics

The modern theory of Integrable Systems and Hydrodynamics are at the origin of a progression of groundbreaking developments and methodologies that are currently deployed to tackle a range of problems in nonlinear science. We provide a broad historical perspective on the subject and a comment on some recent advancements. Particular attention is devoted to prototypical examples aimed at illustrating some features of integrable nonlinear partial differential equations (PDEs) of interest in Hydrodynamics. We show how asymptotic methods and multi-scale techniques are effective for the derivation of integrable model equations for the description of a uid where dissipation and dispersion are negligible or small. Among the possible notions of integrability for a nonlinear PDE, we adopt a definition based on the existence of infinitely many symmetries. This definition is sufficiently general and versatile as it consistently applies to a broad range of hydrodynamic models described by quasilinear, dispersive and viscous PDEs.