| **Proceedings of the Conference on Nonlinear Coherent Structures in Physics and Biology,** Heriot-Watt University, Edinburgh, July 95 | |

*Asymptotic Integrability Of Water Waves*

### A.S. Fokas

Department of Mathematical Sciences,

Loughborough University of Technology,

Loughborough, Leics, LE11 3TU, UK

Abstract

It was established 100 years ago by Korteweg and de Vries that unidirectional
idealized water waves are asymptotically integrable to $0(\epsilon)$, where
$\epsilon = \frac{a}{h_0} = \frac{h_0^2}{\ell^2}$, $a$ and $\ell$ are typical
values of the amplitude and of the wavelength of the waves, and $h_0$ is the
undisturbed depth of the water. Kodama has extended the asymptotic
integrability of this system to $0(\epsilon^2)$: He has found an explicit
transformation which maps this system to the integrable equation describing
the next commuting flow of the kdV hierarchy. We have recently:
Presented a generalization of Kodama's transformation which maps this
system to KdV equation itself. Furthermore, we have shown that this system can
be mapped to certain other integrable equations.

Established the asymptotic integrability of the idealized water waves to
$0(\epsilon^2)$ without the unidirectionality asymption.

The concept of the mastersymmetry plays a crucial role in the derivation of
the above results.

Conference paper.

*Last modified Mon Apr 8 16:24:54 GB-Eire 1996
(DBD)*