  We construct solitary wave solutions in a $1+1$ dimensional massless scalar
($\phi$) field theory with a specially chosen potential $V(\phi)$. The equation
governing perturbations about this solitary wave has an effective potential
which is a simple harmonic well over a region, and a constant beyond. This
feature allows us to ensure the stability of the solitary wave through the
existence of bound states in the well, which can be found by semi-analytical
methods. A further check on stability is performed through our search for
quasi-normal modes (QNM) which are defined for purely outgoing boundary
conditions. The time-domain profiles of the perturbations and the parametric
variation of the QNM values are presented and discussed in some detail.
Expectedly, a damped oscillatory temporal behaviour (ringdown) of the
fluctuations, at finite values of the spatial variable $x$, is clearly seen
through our analysis of the quasi-normal modes.
