Focusing mKdV equation: Two-phase solutions and their stability analysis

Abstract

In this work, we primarily focus on the two-phase solutions and their stability to the focusing mKdV equation. By employing the algebro-geometric approach in combination with an effective integration method, we construct explicit two-phase solutions and their corresponding wave-functions expressed in terms of the Riemann theta function. The spectral stability of two-phase solutions is examined via a modified squared-eigenfunction approach, and their stability with respect to subharmonic perturbations is further analyzed under spectrally unstable conditions. In addition, the orbital stability of the two-phase solutions is investigated. To the best of our knowledge, this study provides the first rigorous stability theory for the two-phase solutions of the focusing mKdV equation.