  In this paper we study the stability problem for KdV solitons on the left and
right half-line. Unlike standard KdV, these are not exact solutions to the
equations posed in the half-line. However, we are able to show that solitons
placed initially far away from the origin are strongly stable for the problem
posed on the right half-line, assuming homogeneous boundary conditions. For the
problem posed on the left half-line, the positive infinitetime stability
problem makes no sense for the case of KdV solitons, but in this setting we
prove a result of stability for all negative times. The proof involves the use
of almost conserved quantities adapted to the evolution of the KdV soliton on
the particular case of the half-line. Adaptations to other boundary conditions
or star graphs are also discussed.
