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Regularized long-wave expansion

Though the long-wave expansion is expected to converge for $W \to +\infty$ with $R$ kept finite, it is quite doubtful whether it converges for finite $W$ and finite $R$. 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

\begin{displaymath}
\partial_t h + \partial_x h + \partial_x ^2 h + \partial_x ^3 h + \partial_x ^4 h + \cdots + \partial_x ^n h + \cdots
= 0
\end{displaymath} (6)

which contains, formally, an infinite number of terms. Since $\partial_x $ is not a bounded operator, the left-hand side diverges and therefore it seems meaningless. However, by letting $-\partial_x $ 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 $\partial_x =0$ 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 $A^{(1)} = A^{(1)}(h)$ and $A^{(2)} = A^{(2)}(h)$ we define $\hat{L} \stackrel{\mathrm{def}}{=}1 + A^{(1)}\partial_x + A^{(2)}\partial_x ^2$ and thereby

\begin{displaymath}
S \stackrel{\mathrm{def}}{=}\hat{L}Q
= S_0 + \mu S_1 + \mu^2 S_2 + \cdots
;
\end{displaymath} (7)

by remembering $\partial_x =\mu\partial_{x_{1}}$, we have

\begin{eqnarray*}
S_0 = Q_0, \qquad
S_1 = Q_1 + A^{(1)}\partial_{x_{1}}{Q_0}, ...
...l_{x_{1}}{Q_1} + A^{(2)}\partial_{x_{1}}^2{Q_0}\ \ \hspace*{5cm}
\end{eqnarray*}



and so on. According to the philosophy of Pade' approximation, $A^{(1)}$ and $A^{(2)}$ are determined so that ``$S_2$ may vanish'' which constitutes the regularization of the long-wave expansion (4). The result is
\begin{displaymath}
A^{(1)} = -\frac{20}{21}Rh^4, \quad
A^{(2)} = -h^2
\end{displaymath} (8)

which leads to the regularized equation
$\displaystyle \partial_t {h} - \frac{4}{21} R \partial_x \partial_t (h^5) - \partial_x \left( h^2\partial_x \partial_t {h} \right)
\hspace*{8em}$      
$\displaystyle {}
+ \frac{2}{3}\partial_x \left[{
h^3 - \partial_x \left( \frac{\cot\alpha }{4}h^4 + \frac{72}{245}Rh^7 \right)
+ W h^3\partial_x ^3{h}
}\right]$ $\textstyle =$ $\displaystyle 0. \hspace{2em}$ (9)

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 $\delta_* \stackrel{\mathrm{def}}{=}R/W^{1/3} \lesssim 1$, Eq. (9) is quantitatively valid as well.

\includegraphics{bifurc.ps}



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Author: OOSHIDA Takeshi
ooshida@damp.tottori-u.ac.jp 2000-08-24