  We study an analytically tractable model with long-range interactions for
which an out-of-equilibrium very long-lived coherent structure spontaneously
appears. The dynamics of this model is indeed very peculiar: a bicluster forms
at low energy and is stable for very long time, contrary to statistical
mechanics predictions. We first explain the onset of the structure, by
approximating the short time dynamics with a forced Burgers equation. The
emergence of the bicluster is the signature of the shock waves present in the
associated hydrodynamical equations. The striking quantitative agreement with
the dynamics of the particles fully confirms this procedure. We then show that
a very fast timescale can be singled out from a slower motion. This enables us
to use an adiabatic approximation to derive an effective Hamiltonian that
describes very well the long time dynamics. We then get an explanation of the
very long time stability of the bicluster: this out-of-equilibrium state
corresponds to a statistical equilibrium of an effective mean-field dynamics.
