  In this paper we give a systematic and simple account that put in evidence
that many breather solutions of integrable equations satisfy suitable
variational elliptic equations, which also implies that the stability problem
reduces in some sense to $(i)$ the study of the spectrum of explicit linear
systems (\emph{spectral stability}), and $(ii)$ the understanding of how bad
directions (if any) can be controlled using low regularity conservation laws.
We exemplify this idea in the case of the modified Korteweg-de Vries (mKdV),
Gardner, and sine-Gordon (SG) equations. Then we perform numerical simulations
that confirm, at the level of the spectral problem, our previous rigorous
results, where we showed that mKdV breathers are $H^2$ and $H^1$ stable,
respectively. In a second step, we also discuss the Gardner and the Sine-Gordon
cases, where the spectral study of a fourth-order linear matrix system is the
key element to show stability. Using numerical methods, we confirm that all
spectral assumptions leading to the $H^2\times H^1$ stability of SG breathers
are numerically satisfied, even in the ultra-relativistic, singular regime. In
a second part, we study the periodic mKdV case, where a periodic breather is
known from the work of Kevrekidis et al. We rigorously show that these
breathers satisfy a suitable elliptic equation, and we also show numerical
spectral stability. However, we also identify the source of nonlinear
instability in the case described in Kevrekidis et al. Finally, we present a
new class of breather solution for mKdV, believed to exist from geometric
considerations, and which is periodic in time and space, but has nonzero mean,
unlike standard breathers.
