Transition from stable to unstable growth by an inertial force

We introduce a simple growth model where the growth of the interface is affected by an inertial force and a white noise. The magnitude of the inertial force is controlled by a constant p between 0 and 1. An inertial force increases continuously from 0, as p does from 0 to 1. In our model, the interface starts growing from a flat state. When p<p(c), the interface width in our model increases continuously from 0 as time elapses, but it saturates to a constant value in the long time limit. The saturated values of the interface width are the same for different values of p if p<p(c). When p>p(c), however, the interface width increases continuously without saturation as time elapses. We explain via simple calculation how this interesting phenomenon occurs in our model. We find p(c)=0.5 from the calculation. This critical value is in excellent agreement with the critical value p(c)=0.50(1) found from the simulations of our model.