  We investigate a novel mapping between solutions to several members of the
Klein-Gordon family of equations and solutions to equations describing their
reductions via the slowly varying envelope approximation. This mapping creates
a link between the study of interacting relativistic fields and that of systems
more amenable to laboratory-based analogue research, the latter described by
nonlinear Schr\"odinger equations. A new evolution equation is also derived,
emerging naturally from the sine-Gordon formula, possessing a Bessel-function
nonlinearity; numerical investigations show that solutions to this novel
equation include quasi-solitary waves, breather solutions, along with pulse
splittings and recombinations.
