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We provide a set of exact solutions in field theory of scalar fields with$Z_2$ symmetry that involve Jacobi elliptic functions. These solutions have theinteresting property to provide massive waves even if one starts from amassless equation. We analyze them classically providing also exact solutionsto the corresponding equations for the Green functions needed to completelysolve them in a strong coupling limit. This is accomplished using a functionalexpansion into powers of the current. It is shown that the spontaneous breakingof the $Z_2$ symmetry is due to the existence of a zero mode that persists alsoin the case of the Higgs mechanism. In this latter case, the zero mode seems toplay a role similar to the Goldstone boson in the breaking of a continuoussymmetry and so, it should be important at lower momenta as a long rangeexcitation.

Exact solutions and zero modes in scalar field theory