  When floating on a two-dimensional (2D) surface of superfluid $^{4}$He,
electrons arrange themselves in 2D crystalline structure known as Wigner
crystal. In channels, the boundaries interfere the crystalline order and in
case of very narrow channels one observes a quasi-1D Wigner crystal formed by
just a few rows of electrons and, ultimately, one row, i.e., in the ``quantum
wire'' regime. Recently, the ``quantum wire'' regime was accessed
experimentally [D. Rees {\it et al.}, Phys. Rev. Lett. {\bf 108}, 176801
(2012)] resulting in unusual transport phenomena such as, e.g., oscillations in
the electron conductance. Using molecular dynamics simulations, we study the
nonlinear transport of electrons in channels with constrictions: single and
multiple symmetric and asymmetric geometrical constrictions with varying width
and length, and saddle-point-type potentials with varying gate voltage. We
analyze the average particle velocity versus driving force or the gate voltage.
We have revealed a significant difference in the dynamics for long and short
constrictions: the oscillations of the average particle velocity for the
systems with short constrictions exhibit a clear correlation with the
transitions between the states with different numbers of rows of particles
while for longer constrictions these oscillations are suppressed. The obtained
results are in agreement with the experimental observations. We proposed a
FET-like structure that consists of a channel with asymmetric constrictions. We
show that applying a transverse bias results either in increase of the average
particle velocity or in its suppression thus allowing a flexible control tool
over the electron transport. Our results bring important insights into the
dynamics of electrons floating on the surface of superfluid $^{4}$He in
channels with constrictions and allow an effective control over the electron
transport.
