Preprint A332/2004
Asymptotic nonlinear wave modeling through the Dirichlet-to-Neumann operator

Andre Nachbin | Artiles , William

**Keywords: **

New nonlinear evolution equations are derived that generalize the
system by Matsuno and a terrain-following
Boussinesq system by Nachbin.
The regime considers finite-amplitude surface gravity waves on a
two-dimensional incompressible and inviscid fluid of, highly variable, finite
depth. The asymptotic simplification of the nonlinear potential theory
equations is performed through a perturbation anaylsis of the
Dirichlet-to-Neumann operator on a highly corrugated strip. This is
achieved through the use of a curvilinear coordinate system. Rather than
doing a long wave expansion for the velocity potential,
a Fourier-type operator is expanded in a wave steepness parameter.
The novelty is that the topography can vary on a broad range of
scales. It can also have a complex profile including
that of a multiply-valued function. The resulting evolution equations
are variable coefficient Boussinesq-type equations.
These equations represent a fully dispersive system in the sense that
the original (hyperbolic tangent) dispersion relation is not
truncated. The formulation
is done over a periodically extended
domain so that, as an application, it produces efficient Fourier (FFT) solvers.
A preliminary communication of this work has been published in the Physical
Review Letters.