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

## On the Integrability of a New Class of Water Wave Equations

### V. Marinakis ¹, T.C. Bountis

¹Department of Mathematics,
University of Patras,
26110 Patras, Greece

### Abstract

We examine two nonlinear partial differential equations (PDEs) from the point of view of their integrability. The first one is integrable and can be derived from the second equation through a local transformation recently introduced by Fokas \cite{fo2}. Reductions of the first equation of the form $u=u(x)$ or $u=u(x-t)$ involve algebraic singularities but can be shown analytically (Painlev\'e analysis, Goriely transformations) and numerically (ATOMFT algorithm) to be integrable. Moreover, the Painlev\'e analysis of the first PDE leads to a truncated expansion from which the Lax Pair can be obtained. We then make the reduction $\eta=\eta(x)$ for the second PDE and examine the ordinary differential equation (ODE) that we obtain. The Painlev\'e analysis identifies new integrable cases for this ODE which need to be further checked in terms of their implication for integrability of the PDE. However, the violation of the Painlev\'e property in other cases together with a numerical study of the solutions in the complex $x$--plane give strong indications that, in general, this ODE and therefore the second PDE from which it is obtained are not integrable.
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