International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 63, Pages 3979-3993

Approximate solution for Euler equations of stratified water via numerical solution of coupled KdV system

A. A. Halim,1 S. P. Kshevetskii,2 and S. B. Leble1

1Theoretical and Mathematical Physics Department, Gdansk University of Technology, Gdansk 80-952, Poland
2Theoretical Physics Department, Kaliningrad State University, Kaliningrad 236041, Russia

Received 23 December 2002

Copyright © 2003 A. A. Halim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider Euler equations with stratified background state that is valid for internal water waves. The solution of the initial-boundary problem for Boussinesq approximation in the waveguide mode is presented in terms of the stream function. The orthogonal eigenfunctions describe a vertical shape of the internal wave modes and satisfy a Sturm-Liouville problem. The horizontal profile is defined by a coupled KdV system which is numerically solved via a finite-difference scheme for which we prove the convergence and stability. Together with the solution of the Sturm-Liouville problem, the stream functions give the internal waves profile.