A generalization of the classical concept of characteristic for partial differential equations (PDE) is given in the framework of the geometric formal theory for PDE's. In particular, it is given a relation between singularities of Cauchy data and characteristics in order to obtain integral manifolds (solutions) generated by means of characteristics. In this direction it is shown that to any PDE we can associate a "dual" equation having the same characteristics. These equations can be related by means of a sort of B\"acklund transformation. Furthermore, a criterion that relates characteristics and integral cobordism (or quantum cobordism as introduced by A. Prastaro) is given. Al relation between quantum cobordism in non-linear PDE's and Green's functions is given too.
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