Communications in Mathematical Sciences

Volume 16 (2018)

Number 3

A family of asymptotic models for internal waves propagating in intermediate/deep water

Pages: 809 – 819



Daniel G. Alfaro Vigo (Departamento de Ciência da Computação, Universidade Federal do Rio de Janeiro, RJ, Brazil)

Gladys Calle Cardeña (Instituto de Ciência, Engenharia e Tecnologia, Universidade Federal dos Vales de Jequitinhonha e Mucuri, Teófilo Otoni, MG, Brazil)


In this paper, we obtain a family of approximate systems of two partial differential equations for the modeling of weakly nonlinear long internal waves propagating at the interface between two immiscible and irrotational fluids in a channel of intermediate/infinite depth. These systems are approximations of the system of Euler equations that share the same asymptotic order.

The analysis of the corresponding linearized systems leads to the identification of several subfamilies (associated with different subsets in the space of parameters) for which the solutions of the linearized models are physically compatible with the solutions of the linearized system of Euler equations. Finally, for the class of weakly dispersive nonlinear systems which is formed by some of those subfamilies, we establish the existence and uniqueness of local in time solutions.


internal waves models, nonlinear waves equations, dispersive waves equations, local well-posedness

2010 Mathematics Subject Classification

35E15, 35Q35, 35S30

The second author was partially supported by CNPq/Brazil and CAPES/Brazil.

Received 28 October 2016

Received revised 17 February 2018

Accepted 17 February 2018

Published 30 August 2018