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Though the long-wave expansion is expected to converge
for
with
kept finite,
it is quite doubtful
whether it converges for finite
and finite
.
The failure shown by Salamon et al.
seems to suggest that
the long-wave expansion is poorly convergent.
As a model of the ``poor convergence'',
let us consider a linear PDE
 |
(6) |
which contains, formally, an infinite number of terms.
Since
is not a bounded operator,
the left-hand side diverges and therefore it seems meaningless.
However, by letting
operate upon Eq. (6)
and then adding the result to Eq. (6),
we find a PDE containing only a finite number of terms!
The idea is to regard Benney's long-wave expansion
as Taylor expansion around
and then replace it by Pade' approximation.
This procedure can be applied to Eq. (4)
which is a nonlinear PDE[6].
In terms of undetermined functions
and
we define
and thereby
 |
(7) |
by remembering
, we have
and so on.
According to the philosophy of Pade' approximation,
and
are determined so that ``
may vanish''
which constitutes
the regularization of the long-wave expansion (4).
The result is
 |
(8) |
which leads to the regularized equation
Numerical solutions of Eq. (9)
are compared with those of two long-wave equations (see figure).
The failure is successfully avoided,
at least qualitatively.
It is found
that, for
,
Eq. (9) is quantitatively valid as well.
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Author: OOSHIDA Takeshi
ooshida@damp.tottori-u.ac.jp
2000-08-24