American Institute of Mathematical Sciences

July  2012, 11(4): 1497-1522. doi: 10.3934/cpaa.2012.11.1497

A selection of nonlinear problems in water waves, analysed by perturbation-parameter techniques

 1 School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom

Received  June 2011 Revised  September 2011 Published  January 2012

The methods of analysis based on asymptotic expansions, with a small parameter, are briefly outlined. These techniques are then applied to three examples in the theory of water waves, the aim being to demonstrate the effectiveness of this approach. Throughout, we relate this procedure to more rigorous methods.
The classical problem of water waves is formulated, and then various parameter choices are incorporated. The first problem examines the properties of small-amplitude, periodic waves over constant vorticity and, invoking the ideas of breakdown, scaling and matching, the possibility of stagnation is investigated. In the second case, a Camassa-Holm equation is derived; this is found to be relevant only for the horizontal velocity component in the flow at a specific depth. However, this special, integrable equation requires the retention of a number of terms of different asymptotic order--which is the least satisfactory way to use these methods. The final example, which shows how some new aspects of a familiar problem can be obtained by these methods, develops an asymptotic description of edge waves. This leads to a single equation that captures all aspects of the run-up pattern at a shoreline.
Citation: R. S. Johnson. A selection of nonlinear problems in water waves, analysed by perturbation-parameter techniques. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1497-1522. doi: 10.3934/cpaa.2012.11.1497
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