The breather of the +4 field theory decays by radiation to infinity. The concept of a solitary wave is still useful, however, because a, the amplitude of the 'far field' radiation, is exponentially small in E , the breather amplitude. (The phrase 'weakly non-local' in the title means that the quasisoliton has non-zero but very tiny amplitude as 1x1m.) In this paper, we introduce novel numerical methods to compute G4 breathers. We calculate solutions both on a finite, spatially periodic interval

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... and on We find that spatially periodic breathers are permanent and non-decaying. If we relax the requirement that the soliton decay exponentially for large 1x1, we may also compute non-decaying, weakly non-local breathers on the infinite interval, too. The latter computation requires a mixed spectral basis of rational Chebyshev functions and a special 'radiation function'. We show it is easy to modify these boundary value solutions to mimic radiatively decaying breathers. These techniques make it possible, for the first time, to directly study the Q4 breather for large amplitude. More important, these methods can be applied to other weakly radiating, quasisolitary waves. The concept of the soliton no longer need be, limited to waves that are strictly localised and immortal. AMS classification scheme numbers: 81C05, 65N35, 42C10 x E [-m, a].