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Failure of Benney's long-wave expansion

Salamon et al.[5] obtained permanent solutions to Eqs. (1)-(3) by DNS, and compared them with the corresponding results of the long-wave expansion. It then turned out that Eq. (5) makes false predictions for most part of the bifurcation diagram. In addition, this failure was not improved by proceeding to higher-order expansion.

A remarkable misprediction of Eq. (5) is that for $R>R_*$ permanent solutions cease to exist, in contradiction to experiments; see the $\times$'s in the figure. Non-steady numerical solutions of Eq. (5) for $R>R_*$ is known to diverge in a finite time[4], which also contradicts experimental observations. In short, Benney's long-wave method loses validity when $R\gtrsim {R_*}$.




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