Journal of Applied Mathematics
Volume 2012 (2012), Article ID 910659, 24 pages
http://dx.doi.org/10.1155/2012/910659
Research Article

Step Soliton Generalized Solutions of the Shallow Water Equations

A. C. Alvarez,1,2 A. Meril,3 and B. Valiño-Alonso4

1Department of Fluid Dynamic, IMPA, Dona Castorina 110, Jardín Botànico, 22460-320 Rio de Janeiro, RJ, Brazil
2Oceanology Institute, Environmental Agency, Avenida Primera, 18406, Flores, Playa, 11600 C. Habana, Cuba
3Laboratoire AOC, Université des Antilles et de la Guyane, Campus de Fouillole, 97159 Pointe-à-Pitre, Guadeloupe
4Departamento de Matemáticas, Facultad de Matemáticas y Computación, Universidad de la Habana, San Lázaro esq. A L, 10400 La Habana, Cuba

Received 5 January 2012; Revised 11 April 2012; Accepted 19 April 2012

Academic Editor: Armin Troesch

Copyright © 2012 A. C. Alvarez 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.

Abstract

Generalized solutions of the shallow water equations are obtained. One studies the particular case of a generalized soliton function passing by a variable bottom. We consider a case of discontinuity in bottom depth. We assume that the surface elevation is given by a step soliton which is defined using generalized solutions (Colombeau 1993). Finally, a system of functional equations is obtained where the amplitudes and celerity of wave are the unknown parameters. Numerical results are presented showing that the generalized solution produces good results having physical sense.