A Problem in the Classical Theory of Water Waves: Weakly Nonlinear Waves in the Presence of Vorticity

DOI

10.1142/S1402925112400128How to use a DOI?

Keywords

Water waves; soliton equations; asymptotic expansions; nonlinear waves; vorticity

Abstract

The classical water-wave problem is described, and two parameters (ε-amplitude; δ-long wave or shallow water) are introduced. We describe various nonlinear problems involving weak nonlinearity (ε → 0) associated with equations of integrable type (“soliton” equations), but with vorticity. The familiar problem of propagation governed by the Korteweg–de Vries (KdV) equation is introduced, but allowing for an arbitrary distribution of vorticity. The effect of the constant vorticity on the solitary wave is described. The corresponding problem for the Nonlinear Schrödinger (NLS) equation is briefly mentioned but not explored here. The problem of two-way propagation (admitting head-on collisions), as described by the Boussinesq equation, is examined next. This leads to a new equation: the Boussinesq-type equation valid for constant vorticity. However, this cannot be transformed into an integrable Boussinesq equation (as is possible for the corresponding KdV and NLS equations). The solitary-wave solution for this new equation is presented. A description of the Camassa–Holm equation for water waves, with constant vorticity, with its solitary-wave solution, is described. Finally, we outline the problem of propagation of small-amplitude, large-radius ring waves over a flow with vorticity (representing a background flow in one direction). Some properties of this flow, for constant vorticity, are described.

Copyright

© 2012 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Robin Stanley Johnson
PY  - 2012
DA  - 2012/11/28
TI  - A Problem in the Classical Theory of Water Waves: Weakly Nonlinear Waves in the Presence of Vorticity
JO  - Journal of Nonlinear Mathematical Physics
SP  - 137
EP  - 160
VL  - 19
IS  - Supplement 1
SN  - 1776-0852
UR  - https://doi.org/10.1142/S1402925112400128
DO  - 10.1142/S1402925112400128
ID  - Johnson2012
ER  -