On the existence, uniqueness and stability of Periodic Waves for the fractional Benjamin-Bona-Mahony equation

TL;DR

This paper investigates the existence, uniqueness, and spectral stability of periodic traveling waves in the fractional Benjamin-Bona-Mahony equation, providing theoretical conditions and numerical methods for stability analysis.

Contribution

It establishes sufficient conditions for uniqueness and spectral stability of periodic waves and introduces Petviashvili's method for numerical stability investigation.

Findings

Unique single-lobe solutions under certain conditions

Spectral stability confirmed by eigenvalue analysis

Numerical verification using Petviashvili's method

Abstract

The existence, uniqueness and stability of periodic traveling waves for the fractional Benjamin-Bona-Mahony equation is considered. In our approach, we give sufficient conditions to prove a uniqueness result for the single-lobe solution obtained by a constrained minimization problem. The spectral stability is then shown by determining that the associated linearized operator around the wave restricted to the orthogonal of the tangent space related to the momentum and mass at the periodic wave has no negative eigenvalues. We propose the Petviashvili's method to investigate the spectral stability of the periodic waves for the fractional Benjamin-Bona-Mahony equation, numerically. Some remarks concerning the orbital stability of periodic traveling waves are also presented.