  This paper investigates the non-linear self-interaction of quadrupole
gravitational waves generated by an isolated system. The vacuum Einstein field
equations are integrated in the region exterior to the system by means of a
post-Minkowskian algorithm. Specializing in the quadrupole-quadrupole
interaction (at the quadratic non-linear order), we recover the known results
concerning the non-local modification of the ADM mass-energy of the system
accounting for the emission of quadrupole waves, and the non-local memory
effect due to the re-radiation by the stress-energy distribution of linear
waves. Then we compute all the local (instantaneous) terms which are associated
in the quadrupole-quadrupole metric with the latter non-local effects.
Expanding the metric at large distances from the system, we obtain the
corresponding radiation-field observables, including all non-local and
transient contributions. This permits notably the completion of the observable
quadrupole moment at the 5/2 post-Newtonian order.
