  Discussed in the present study are details of the behavior of low-frequency
"acoustic" ($ac$) modes in the spectrum of edge magnetoplasma oscillations in
axially symmetric degenerate 2D electron systems where electron density
distribution $n(\vec r)$ behaves as $n(\vec r \to R)\to 0$ when $r$ approaches
the external radius $R$ of the domain occupied by electrons. It is shown that
finding the dependence of the spectrum of $ac$-modes on radial ($l$) and
azimuthal ($m$) indices when both $l$ and $m$ are small requires axially
symmetric solution of the relevant problem. The desires is achieved in the
so-called elliptic approximation for electron density distribution $n(\vec r)$.
The obtained results are employed to interpret available data on the excitation
of $ac$-modes in degenerate electron disks with smooth electron density profile
placed in the magnetic field $H$ normal to the disk plane. The performed
analysis confirms reported detection of the soft $ac$-mode in the range of $H
\gg H_{max}$ where $H_{max}$ is the field at which the maximum of the
$\omega_{lm}^{s}(H)$ curve is observed (here and below $\omega_{lm}^{s}(H)$
stands for the frequency of the soft $ac$-mode).
  A strong interaction of the $ac$-modes with $(integer)$-channels inevitably
arising near the boundaries of 2D electron systems with smooth electron density
profile as the magnetic field is varied in the Quantum Hall Effect (QHE) regime
is emphasized. Well-formed $integer$-stripes can suppress some acoustic modes
which is actually observed in experiments.

Mesoscale and Nanoscale Physics