Spectral Stability of Periodic Traveling Wave Solutions for a Double Dispersion Equation

TL;DR

This paper analyzes the spectral stability of periodic traveling wave solutions in a double dispersion equation, providing explicit solutions, characterizations, and stability criteria using Floquet theory and spectral analysis.

Contribution

It introduces explicit periodic wave solutions and characterizes all positive solutions, applying Floquet theory to analyze spectral stability in a double dispersion model.

Findings

Explicit periodic wave solutions derived

Monotonicity of the period map established

Spectral stability proven through eigenvalue analysis

Abstract

In this paper, we investigate the spectral stability of periodic traveling waves for a cubic-quintic and double dispersion equation. Using the quadrature method we find explict periodic waves and we also present a characterization for all positive and periodic solutions for the model using the monotonicity of the period map in terms of the energy levels. The monotonicity of the period map is also useful to obtain the quantity and multiplicity of non-positive eigenvalues for the associated linearized operator and to do so, we use tools of the Floquet theory. Finally, we prove the spectral stability by analysing the difference between the number of negative eigenvalues of a convenient linear operator restricted to the space constituted by zero-mean periodic functions and the number of negative eigenvalues of the matrix formed by the tangent space associated to the low order conserved quantities of the evolution model.