Intermediate long wave systems for internal waves

Abstract

This paper, which deals with internal waves in the two-layer formulation, consists of three parts. The first part is devoted to the derivation of asymptotic models in the intermediate long wave (ILW) regime for internal waves with a free upper surface and with a flat bottom. Using a method similar to that introduced by Bona et al (2008 J. Math. Pures Appl. 89 538–66), we obtain a one-parameter ILW system which is consistent with the full Euler system. In the second part we investigate the well-posedness of the ILW system under the rigid lid assumption. The local well-posedness on the time scale in lower order Sobolev spaces and the large time well-posedness on the long time scale in higher order Sobolev spaces are proven for both 1D and 2D cases. The long time existence result seems to be the first of this type in the context of internal waves. In the third part we provide a rigorous justification to show that the Benjamin–Ono (BO) system can be deduced from the ILW system by letting the depth of the lower flow tend to ∞. The convergence results between the solutions of the BO system and the ILW system are established in both the short time scale and the long time scale , in different functional settings.