  We consider the classical problem of existence, uniqueness and asymptotics of
monotone solutions to the travelling wave equation associated to the parabolic
semi-group equation of a super-Brownian motion with a general branching
mechanism. Whilst we are strongly guided by the probabilistic reasoning of
Kyprianou (2004) for branching Brownian motion, the current paper offers a
number of new insights. Our analysis incorporates the role of Seneta-Heyde
norming which, in the current setting, draws on classical work of Grey (1974).
We give a pathwise explanation of Evans' immortal particle picture (the spine
decomposition) which uses the Dynkin-Kuznetsov N-measure as a key ingredient.
Moreover, in the spirit of Neveu's stopping lines we make repeated use of
Dynkin's exit measures. Additional complications arise from the general nature
of the branching mechanism. As a consequence of the analysis we also offer an
exact X(log X)^2 moment dichotomy for the almost sure convergence of the
so-called derivative martingale at its critical parameter to a non-trivial
limit. This differs to the case of branching Brownian motion and branching
random walk where a moment `gap' appears in the necessary and sufficient
conditions.
