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.

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