A researcher at the Laboratory of Information Technologies at JINR Igor Barashenkov and a lecturer at the Department of Mathematics and Applied Mathematics of the University of Cape Town Nora Alexeeva presented a new variational method employing amplitude and width as collective coordinates of the Klein-Gordon oscillon. Earlier methods resulted in dynamical systems with unstable periodic orbits that blow up when perturbed. An essential feature of the proposed trial function is the inclusion of the third collective variable: correction for the nonuniform phase growth. In addition to determining the parameters of the oscillon, the authors’ approach detects the onset of its instability. The work was published in 2023 in the Physical Review D journal.

The study was motivated by the numerous links and similarities between the Klein-Gordon oscillons and solitons of the nonlinear Schrödinger equations. A simple yet powerful approach to the Schrödinger solitons exploits the variation of the action. By contrast, the variational analysis of the Klein-Gordon oscillons has not been as successful.

One of the obstacles to the straightforward (“naive”) variational treatment of the oscillon is that its width proves to be unsuitable as a collective coordinate in this approach. The soliton amplitude and width comprise a standard set of variables in the Schrödinger domain, but making a similar choice in the Klein-Gordon Lagrangian results in a singular four-dimensional system.

The paper presents a variational method free from singularities. The method aims at determining the oscillon parameters, domain of existence, and stability-instability transition points. The proposed formulation is based on a fast harmonic ansatz supplemented by the adiabatic evolution of the oscillon collective coordinates. An essential component of the set of collective coordinates is the “lazy phase”: a cyclic variable accounting for nonuniform phase acquisitions.

The authors used the Kosevich-Kovalev model as a prototype equation exhibiting oscillon solutions. The variational method establishes the domain of existence of the oscillon (0 < ω < 2) and identifies the frequency ω_c at which the oscillon loses its stability (ω_c = √2). The predicted stability domain is in good agreement with numerical simulations of the partial differential equation ϕ_tt−ϕ_xx+4ϕ−2ϕ³ = 0 which yields stable oscillons with frequencies 1.03ω_c ≤ ω < 2.The variational amplitude-frequency and width-frequency curves are consistent with the characteristics of the numerical solutions.