Madelung, Gross-Pitaevskii and Korteweg
R\'emi Carles (I3M), Rapha\"el Danchin (LAMA), Jean-Claude Saut, (LM-Orsay)
TL;DR
This paper reviews the hydrodynamic formulation of the nonlinear Schrödinger equation through the Madelung transform, connecting quantum hydrodynamics with Korteweg-type fluid models.
Contribution
It provides a comprehensive survey of the Madelung transform's role in linking quantum and classical fluid models, highlighting recent developments.
Findings
Clarifies the mathematical connection between quantum hydrodynamics and Korteweg fluids.
Summarizes recent advances in the hydrodynamic formulation of nonlinear Schrödinger equations.
Identifies open problems and future research directions in the field.
Abstract
This paper surveys various aspects of the hydrodynamic formulation of the nonlinear Schrodinger equation obtained via the Madelung transform in connexion to models of quantum hydrodynamics and to compressible fluids of the Korteweg type.
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