## 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.

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 language | English |
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Title of host publication | Encyclopedia of Mathematical Physics |

Subtitle of host publication | 2nd Edition |

Publisher | Elsevier |

Publication status | Accepted/In press - 23 Dec 2023 |