On the Stability of Periodic Solutions of the Generalized Benjamin-Bona-Mahony Equation

Abstract

We study the stability of a four parameter family of spatially periodic traveling wave solutions of the generalized Benjamin-Bona-Mahony equation to two classes of perturbations: periodic perturbations with the same periodic structure as the underlying wave, and long-wavelength localized perturbations. In particular, we derive necessary conditions for spectral instability to perturbations to both classes of perturbations by deriving appropriate asymptotic expansions of the periodic Evans function, and we outline a nonlinear stability theory to periodic perturbations based on variational methods which effectively extends our periodic spectral stability results.

arXiv comment27 pages, 3 figures

References in corpus (3)

• On the Modulation Equations and Stability of Periodic GKdV Waves via Bloch Decompositions
• An index theorem for the stability of periodic traveling waves of KdV type
• Nonlinear Stability of Periodic Travelling Wave Solutions for the Regularized Benjamin-Ono and BBM Equations

Cited by in corpus (5)

• On the Modulation Equations and Stability of Periodic GKdV Waves via Bloch Decompositions
• Modulational Instability in Equations of KdV Type
• Orbital stability of one-parameter periodic traveling waves for dispersive equations and applications
• An Instability Index Theory for Quadratic Pencils and Applications
• Rigorous Justification of the Whitham Modulation Equations for the Generalized Korteweg-de Vries Equation