  The 1D bump-on-tail problem is studied in order to determine the influence of
drag on quasi-steady solutions near marginal stability ($1-\gamma_d/\gamma_L\ll
1$) when effective collisions are much larger than the instability growth rate
($\nu \gg \gamma$). In this common tokamak regime, it is rigorously shown that
the paradigmatic Berk-Breizman cubic equation for the nonlinear mode evolution
reduces to a much simpler differential equation, dubbed the time-local cubic
equation, which can be solved directly. It is found that in addition to
increasing the saturation amplitude, drag introduces a shift in the apparent
oscillation frequency by modulating the saturated wave envelope. Excellent
agreement is found between the analytic solution for the mode evolution and
both the numerically integrated Berk-Breizman cubic equation and fully
nonlinear 1D Vlasov simulations. Experimentally isolating the contribution of
drag to the saturated mode amplitude for verification purposes is explored but
complicated by the reality that the amount of drag can not be varied
independently of other key parameters in realistic scenarios. While the effect
of drag is modest when the ratio of drag to scattering $\alpha/\nu$ is very
small, it can become substantial when $\alpha/\nu \gtrsim 0.5$, suggesting that
drag should be accounted for in quantitative models of fast-ion-driven
instabilities in fusion plasmas.
