On a Klein-Gordon Reduction for Oscillons

TL;DR

This paper investigates oscillon dynamics in 1D field theories with cubic nonlinearity, using a harmonic reduction to analyze steady states, stability, and their relation to original PDE solutions, combining analytical and numerical methods.

Contribution

It introduces a harmonic reduction approach for oscillons, develops stability criteria, and systematically compares reduced and original models to gain new insights.

Findings

Steady states and stability criteria for the reduced model are established.

Numerical simulations confirm the predictions of the reduced model.

Connections between reduced and original PDE oscillons are elucidated.

Abstract

In the present work we examine the dynamics of a model for oscillons in 1-dimensional spacetime field theories with a cubic nonlinearity. We utilize a reduction of the model to first and third harmonics, which leads to a reduced partial differential equation (PDE) system whose steady states are candidates for the original PDE oscillons. We analyze the steady states of this model and their stability, including via tools such as index theory. We develop suitable functionals needed for the study of such stationary states, as well as an analogue of the famous Vakhitov-Kolokolov criterion for a quantity whose change of monotonicity reflects a change of stability. Then, we test the relevant predictions, over the full range of oscillon frequencies, through systematic numerical computations of both the reduced model, its steady states and stability, and also of the original PDE model, identifying its time-periodic oscillon solution. Our results yield some significant connections with previous studies, but also some fundamental new insights both on the reduced system and the dynamics of the original system.