Periodic traveling waves in the \phi^4 model: Instability, stability and localized structures

TL;DR

This paper numerically investigates the stability of periodic solutions in the ^4 model, revealing conditions under which waves are stable or unstable, and exploring the evolution of unstable states with localized structures.

Contribution

It provides a detailed numerical analysis of the stability properties of dnoidal, cnoidal, and snoidal waves in the ^4 model, including bifurcation and dynamical evolution insights.

Findings

Dnoidal waves are modulationally unstable in the superluminal regime.

Cnoidal waves can be modulationally stable with spectral bands along the imaginary axis.

Snoidal waves are spectrally unstable under subharmonic perturbations in the subluminal regime.

Abstract

We consider the instability and stability of periodic stationary solutions to the classical \phi^4 equation numerically. In the superluminal regime, the model possesses dnoidal and cnoidal waves. The former are modulationally unstable and the spectrum forms a figure eight intersecting at the origin of the spectral plane. The latter can be modulationally stable and the spectrum near the origin in that case is represented by vertical bands along the purely imaginary axis. The instability of the cnoidal states in that case stems from elliptical bands of complex eigenvalues far from the spectral plane origin. In the subluminal regime, there exist only snoidal waves which are modulationally unstable. Considering the subharmonic perturbations, we show that the snoidal waves in the subluminal regime are spectrally unstable with respect to all subharmonic perturbations, while for the dnoidal and cnoidal waves in the superluminal regime, the transition between the spectrally stable state and the spectrally unstable state occurs through a Hamiltonian Hopf bifurcation. The dynamical evolution of the unstable states is also considered, leading to some interesting spatio-temporal localization events.