  By using the scheme of Jacobi elliptic functions with their duality
symmetries we present a formulation of the Jacobi- Gordon field theory that
will manifest the strong/weak coupling duality at classical level; for certain
continuous limits for the elliptic modulus the model will reduce to the
standard sin/sinh Gordon field theories, for which such a strong/weak duality
is known only at the level of the S-matrix. It is shown that the so called
self-dual point for the standard sin/sinh Gordon field theory that divides the
strong and the weak coupling regimes, corresponds only to one point of a set of
fixed points under the duality transformations for the elliptic functions. The
potentials constructed in terms of elliptic functions have a critical behavior
near that self-dual point, showing a change of topology; in the weak coupling
regime the vacuum topology implies that there exists the possibility of
formation of topological defects, and in the strong regime coupling there no
exists the possibility of formation of those defects. Furthermore, the
equations of motion can be solved in exact form in terms of the inverse
elliptic functions; in a case the kink-like solitons asso\-cia\-ted with the
maxima of the potential can decay to cusp-like solitons associated with the
minima. The polynomial expansions of the generalized models show a critical
behavior at certain self-dual points; such points define the regions where the
spontaneous symmetry breaking scenarios are po\-ssi\-ble. By invoking the
duality symmetries for the elliptic functions, an explicit relation between the
original potentials and their dual versions are constructed; with this
relationship,
  an approaching to a specific self-dual point is considered for our
generalized models.
