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 permanent solutions cease to exist,
in contradiction to experiments;
see the
's in the figure.
Non-steady numerical solutions of Eq. (5)
for
is known to diverge in a finite time[4],
which also contradicts experimental observations.
In short, Benney's long-wave method loses validity
when
.