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

Abstract

We consider the instability and stability of periodic stationary solutions to the classical ϕ^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.

To appear in Physical Review E

References in corpus (8)

• The Stability Spectrum for Elliptic Solutions to the Focusing NLS Equation
• Collective coordinate model of kink-antikink collisions in ${\varphi }^4$ theory
• Modulational instability of periodic standing waves in the derivative NLS equation
• The Stability Spectrum for Elliptic Solutions to the Sine-Gordon Equation
• Localized structures on librational and rotational travelling waves in the sine-Gordon equation
• Kink-antikink stripe interactions in the two-dimensional sine-Gordon equation
• Stability of elliptic solutions to the sinh-Gordon equation
• Constrained instantons and kink-antikink collisions