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

Controlling the transport of electrons on superfluid $^{4}$He in symmetric and asymmetric FET-like structures