  Peakons are special weak solutions of a class of nonlinear partial
differential equations modelling non-linear phenomena such as the breakdown of
regularity and the onset of shocks. We show that the natural concept of weak
solutions in the case of the modified Camassa-Holm equation studied in this
paper is dictated by the distributional compatibility of its Lax pair and, as a
result, it differs from the one proposed and used in the literature based on
the concept of weak solutions used for equations of the Burgers type.
Subsequently, we give a complete construction of peakon solutions satisfying
the modified Camassa-Holm equation in the sense of distributions; our approach
is based on solving certain inverse boundary value problem the solution of
which hinges on a combination of classical techniques of analysis involving
Stieltjes' continued fractions and multi-point Pad\'{e} approximations. We
propose sufficient conditions needed to ensure the global existence of peakon
solutions and analyze the large time asymptotic behaviour whose special
features include a formation of pairs of peakons which share asymptotic speeds,
as well as Toda-like sorting property.
