  In this paper we study the stability problem for mKdV breathers on the left
half-line. We are able to show that leftwards moving breathers, initially
located far away from the origin, are strongly stable for the problem posed on
the left half-line, when assuming homogeneous boundary conditions. The proof
involves a Lyapunov functional which is almost conserved by the mKdV flow once
we control some boundary terms which naturally arise.
