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.