It is proved that a system of three-wave coupling with weak damping and the same sign of the coupling coefficients can be completely described by the Weierstrass function. On the basis of the bifurcation theory of the Weierstrass function that we have developed, criteria are found for distinguishing three phases of the three-wave coupling, namely the stable phase and the excited explosive and auto-explosive phases. Two kinds of explosion mechanisms in the three-wave coupling are found and described. The mechanism for the excited explosive phase is normal because trigger energy is required in such a mechanism and the threshold λth is proportional to δ2 (δ is the frequency mismatch), whereas the anomalous mechanism for the auto-explosive phase does not need any trigger – only proper changes of internal parameters to make the absolute invariant J less than 27 are required. In addition, the repetition period of the explosive instability is given. Finally, the effects of weak damping on the bifurcation structure of the explosive instability in three-wave interactions are analysed.