  We study the short-time stability of quantum dynamics in
quasi-one-dimensional systems with respect to small localized perturbations of
the potential. To this end, we address, analytically and numerically, the decay
of the Loschmidt echo (LE) during times short compared to the Ehrenfest time.
We find that the LE is generally a non-monotonous function of time and exhibits
strongly pronounced minima and maxima at the instants of time when the
corresponding classical particle traverses the perturbation region. We also
show that, under general conditions, the envelope decay of the LE is well
approximated by a Gaussian, and we derive explicit analytical formulas for the
corresponding decay time. Finally, we demonstrate that the observed
non-monotonicity of the LE decay is only pertinent to one-dimensional (and,
more generally, quasi-one-dimensional systems), and that the short-time decay
of the LE can be monotonous in higher number of dimensions.
