  Nonlinear polaron-like dynamics of the two-dimensional Wigner solid (WS) on
superfluid $^{4}\mathrm{He}$ are theoretically analyzed in different models and
transport regimes for their similarities and distinctions. The Bragg-Cherenkov
(BC) resonant excitation of surface waves and WS decoupling from surface
dimples were usually considered in terms of a dc transport model. At the same
time, field-velocity characteristics of the WS are measured under ac conditions
and presented for time-averaged quantities. Here the nonlinear equation of
motion of the WS coupled to surface dimples is studied for ac conditions using
two different approaches based on fixing the driving field or the output
current. Both approaches are shown to give similar results for the first
harmonics of major transport properties. In the ac theory, the BC-resonances
for dimple inertia and the momentum relaxation rate have asymmetrical shapes,
which is in contrast with the results of dc models. Even a quite low driving
frequency is shown to affect the amplitude of the BC resonance and decoupling
of the WS. Above the BC threshold, the effective mass of surface dimples as a
function of the velocity amplitude strongly oscillates indicating multiple
recoupling processes.
