Orbital Stability of Periodic Traveling Waves for the "abcd" Boussinesq Systems

TL;DR

This paper investigates the orbital stability of periodic traveling wave solutions in the

Contribution

It introduces new stability results for the

Findings

Existence of solutions depending on Jacobi elliptic functions.

Spectral analysis using Floquet theory.

Orbital stability established via abstract theoretical results.

Abstract

New results concerning the orbital stability of periodic traveling wave solutions for the "abcd" Boussinesq model will be shown in this manuscript. For the existence of solutions, we use basic tools of ordinary differential equations to show that the corresponding periodic wave depends on the Jacobi elliptic function of cnoidal type. The spectral analysis for the associated linearized operator is determined by using some tools concerning the Floquet theory. The orbital stability is then established by applying the abstract results [2] and [14] which give us sufficient conditions to the orbital stability for a general class of evolution equations.