Revista Matemática Iberoamericana

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Volume 34, Issue 1, 2018, pp. 245–304
DOI: 10.4171/RMI/985

Published online: 2018-02-06

Long wave asymptotics for the Euler–Korteweg system

Sylvie Benzoni-Gavage[1] and David Chiron[2]

(1) Université Claude Bernard Lyon 1, Villeurbanne, France
(2) Université Côte d'Azur, Nice, France

The Euler–Korteweg system (EK) is a fairly general nonlinear waves model in mathematical physics that includes in particular the fluid formulation of the NonLinear Schrödinger equation (NLS). Several asymptotic regimes can be considered, regarding the length and the amplitude of waves. The first one is the free wave regime, which yields long acoustic waves of small amplitude. The other regimes describe a single wave or two counter propagating waves emerging from the wave regime. It is shown that in one space dimension those waves are governed either by inviscid Burgers or by Korteweg–de Vries equations, depending on the spatio-temporal and amplitude scalings. In higher dimensions, those waves are found to solve Kadomtsev–Petviashvili equations. Error bounds are provided in all cases. These results extend earlier work on defocussing (NLS) (and more specifically the Gross–Pitaevskii equation), and sheds light on the qualitative behavior of solutions to (EK), which is a highly nonlinear system of PDEs that is much less understood in general than (NLS).

Keywords: Euler–Korteweg system, capillary fluids, Korteweg–de Vries equation, Kadomtsev–Petviashvili equation, weakly transverse Boussinesq system

Benzoni-Gavage Sylvie, Chiron David: Long wave asymptotics for the Euler–Korteweg system. Rev. Mat. Iberoamericana 34 (2018), 245-304. doi: 10.4171/RMI/985