Reduction of nonlinear field theory equations to envelope models: towards a universal understanding of analogues of relativistic systems

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.