Journal of Nonlinear Science

, Volume 12, Issue 4, pp 283–318

Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I: Derivation and Linear Theory

  • Bona
  • Chen
  • Saut
Article

DOI: 10.1007/s00332-002-0466-4

Cite this article as:
Bona, Chen & Saut J. Nonlinear Sci. (2002) 12: 283. doi:10.1007/s00332-002-0466-4

Summary.

Considered herein are a number of variants of the classical Boussinesq system and their higher-order generalizations. Such equations were first derived by Boussinesq to describe the two-way propagation of small-amplitude, long wavelength, gravity waves on the surface of water in a canal. These systems arise also when modeling the propagation of long-crested waves on large lakes or the ocean and in other contexts. Depending on the modeling of dispersion, the resulting system may or may not have a linearization about the rest state which is well posed. Even when well posed, the linearized system may exhibit a lack of conservation of energy that is at odds with its status as an approximation to the Euler equations. In the present script, we derive a four-parameter family of Boussinesq systems from the two-dimensional Euler equations for free-surface flow and formulate criteria to help decide which of these equations one might choose in a given modeling situation. The analysis of the systems according to these criteria is initiated.

Key words. water waves, two-way propagation, Boussinesq systems, local wellposedness, global wellposedness

Copyright information

© Springer-Verlag New York Inc. 2002

Authors and Affiliations

  • Bona
    • 1
  • Chen
    • 2
  • Saut
    • 3
  1. 1.Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607, USAUS
  2. 2.Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA e-mail: chen@math.purdue.eduUS
  3. 3.UMR de Mathématiques, Université de Paris-Sud, Bâtiment 425, 91405 Orsay, FranceFR