Rolling in the Higgs Model and Elliptic Functions

TL;DR

This paper develops an improved approximation method for the Higgs (Duffing) equation in the rolling regime by employing hyperbolic function expansions and elliptic functions, enhancing perturbation theory beyond small oscillations.

Contribution

It introduces a novel approach using hyperbolic functions and elliptic functions to approximate solutions in the rolling regime of nonlinear oscillators like the Higgs equation.

Findings

Hyperbolic function expansions outperform trigonometric ones in the rolling regime.

Elliptic function representations provide accurate approximate solutions.

The method extends perturbation theory applicability to non-small oscillation regimes.

Abstract

Asymptotic methods in nonlinear dynamics are used to improve perturbation theory results in the oscillations regime. However, for some problems of nonlinear dynamics, particularly in the case of Higgs (Duffing) equation and the Friedmann cosmological equations, not only small oscillations regime is of interest but also the regime of rolling (climbing), more precisely the rolling from a top (climbing to a top). In the Friedman cosmology, where the slow rolling regime is often used, the rolling from a top (not necessary slow) is of interest too. In the present work a method for approximate solution to the Higgs equation in the rolling regime is presented. It is shown that in order to improve perturbation theory in the rolling regime turns out to be effective not to use an expansion in trigonometric functions as it is done in case of small oscillations but use expansions in hyperbolic functions instead. This regime is investigated using the representation of the solution in terms of elliptic functions. An accuracy of the corresponding approximation is estimated.