  A general family of peakon equations is introduced, involving two arbitrary
functions of the wave amplitude and the wave gradient. This family contains all
of the known breaking wave equations, including the integrable ones:
Camassa-Holm equation, Degasperis-Procesi equation, Novikov equation, and
FORQ/modified Camassa-Holm equation. One main result is to show that all of the
equations in the general family possess weak solutions given by multi-peakons
which are a linear superposition of peakons with time-dependent amplitudes and
positions. In particular, neither an integrability structure nor a Hamiltonian
structure is needed to derive $N$-peakon weak solutions for arbitrary $N>1$. As
a further result, single peakon travelling-wave solutions are shown to exist
under a simple condition on one of the two arbitrary functions in the general
family of equations, and when this condition fails, generalized single peakon
solutions that have a time-dependent amplitude and a time-dependent speed are
shown to exist. Wave breaking criteria are outlined for this general family of
peakon equations. An interesting generalization of the Camassa-Holm and
FORQ/modified Camassa-Holm equations is obtained by deriving the most general
subfamily of peakon equations that possess the Hamiltonian structure shared by
the Camassa-Holm and FORQ/modified Camassa-Holm equations. Peakon
travelling-wave solutions and their features, including a variational
formulation (minimizer problem), are derived for these generalized equations. A
final main result is that $2$-peakon weak solutions are investigated and shown
to exhibit several novel kinds of behaviour, including the formation of a bound
pair consisting of a peakon and an anti-peakon that have a maximum finite
separation.
