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

TL;DR

This paper establishes a novel mapping between relativistic Klein-Gordon equations and nonlinear Schrödinger envelope models, enabling laboratory analogues of relativistic systems and introducing a new Bessel-function nonlinear evolution equation with diverse wave solutions.

Contribution

It introduces a new mathematical mapping linking relativistic field equations to envelope models and derives a novel Bessel-function nonlinear evolution equation.

Findings

Solutions include quasi-solitary waves and breathers

Numerical results show pulse splitting and recombination

New equation bridges relativistic fields and laboratory models

Abstract

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.