Long wave asymptotics for the Euler-Korteweg system
The Euler-Korteweg system EK is a general nonlinear waves model in mathematical physics. In particular, this includes the fluid formulation of the nonlinear Schrödinger equation (NLS). Different asymptotic regimes can be considered, regarding the length and the amplitude of waves. 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. 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 spatiotemporal and amplitude scalings. In higher dimensions, those waves are shown to solve the Kadomtsev-Petviashvili equations. Error bounds are provided in all cases. These results extend earlier work on defocusing NLS equation, and sheds light on the qualitative behavior of solutions to EK, which is a highly nonlinear system of PDEs much less understood than NLS.
scientific article; zbMATH DE number 6856507