Integrability and Hydrodynamics

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Abstract

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 di erential equations (PDEs) of interest in Hydrodynamics. We
show how asymptotic methods and multi-scale techniques are e ective 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 de nition based on the existence of in nitely many symmetries. This
de nition is su ciently general and versatile as it consistently applies to a broad range of
hydrodynamic models described by quasilinear, dispersive and viscous PDEs.
Original languageEnglish
Title of host publicationEncyclopedia of Mathematical Physics
Subtitle of host publication2nd Edition
PublisherElsevier
Publication statusAccepted/In press - 23 Dec 2023

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