  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 the
interesting property to provide massive waves even if one starts from a
massless equation. We analyze them classically providing also exact solutions
to the corresponding equations for the Green functions needed to completely
solve them in a strong coupling limit. This is accomplished using a functional
expansion into powers of the current. It is shown that the spontaneous breaking
of the $Z_2$ symmetry is due to the existence of a zero mode that persists also
in the case of the Higgs mechanism. In this latter case, the zero mode seems to
play a role similar to the Goldstone boson in the breaking of a continuous
symmetry and so, it should be important at lower momenta as a long range
excitation.
