  The dressing method is a technique to construct new solutions in non-linear
sigma models under the provision of a seed solution. This is analogous to the
use of autoBacklund transformations for systems of the sine-Gordon type. In a
recent work, this method was applied in the sigma model that describes string
propagation on $\mathbb{R} \times \mathrm{S}^2$, using as seeds the elliptic
classical string solutions. Some of the new solutions that emerge reveal
instabilities of their elliptic precursors. The focus of the present work is
the fruitful use of the dressing method in the study of the stability of closed
string solutions. It establishes an equivalence between the dressing method and
the conventional linear stability analysis. More importantly, this equivalence
holds true in the presence of appropriate periodicity conditions that closed
strings must obey. Our investigations point to the direction of the dressing
method being a general tool for the study of the stability of classical string
configurations in the diverse class of symmetric spacetimes.
