Spectra and Bifurcations

TL;DR

This paper explores the spectral properties of non-linear wave equations, emphasizing bifurcations and instability phenomena, with insights from computer simulations and implications for soliton behavior and symmetry breaking.

Contribution

It introduces a spectral framework focusing on bifurcations and instability in non-linear wave equations, linking soliton bifurcations to dynamical symmetry breaking.

Findings

Bifurcations reflect instability phenomena in non-linear wave spectra.

Soliton bifurcations indicate symmetry breaking.

Computer simulations show bifurcations in non-stationary states and soliton splitting.

Abstract

The concept of spectrum for a class of non-linear wave equations is studied. Instead of looking for stability, the key to the spectral structure is found in the instability phenomena (bifurcations). This aspect is best seen in the `classical model' of the non-linear wave mechanics. The solitons (macro-localizations) are a part of the non-linear spectral problem; their bifurcations reflect the dynamical symmetry breaking. The computer simulations suggest that the bifurcations of the asymptotic behaviour occur also for the general, non-stationary states. A~phenomenon of the soliton splitting is observed.