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

TL;DR

The paper addresses the stability of genus-two (two-phase) traveling-wave solutions of the focusing mKdV equation by constructing explicit theta-functional solutions via algebro-geometric methods and analyzing their Lax-pair wavefunctions. It develops a rigorous stability theory using the squared-eigenfunction approach to characterize spectral stability, distinguishing Case 1 (spectrally stable under subharmonic perturbations) from Case 2 (spectrally unstable with a modulational threshold) and then proves orbital stability in appropriate periodic Sobolev spaces. The results demonstrate that genus-two traveling waves with all branch points imaginary (Case 1) are spectrally and orbitally stable (including subharmonic cases), while those with mixed complex-conjugate branch points (Case 2) are spectrally unstable but may exhibit subharmonic stability up to a modulus-dependent bound; this constitutes the first rigorous stability theory for genus-two mKdV two-phase solutions. By linking Riemann theta-function representations to elliptic-function forms and deriving explicit Lax-pair wavefunctions, the work provides a comprehensive framework for stability analysis in higher-genus finite-gap solutions and enhances understanding of nonlinear wave dynamics in integrable systems.

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.

Figures (6)

• Figure 1: The homology basis for the curve $y^2=\prod_{i=1}^{3}\left(\lambda^2-\lambda_i^2\right)$ defined in equation \ref{['eq:define-curve-algebro']}. The \ref{['fig:genus-two-figure-p']} and \ref{['fig:genus-two-figure-c']} corresponds to the \ref{['case1']} and \ref{['case2']}, respectively.
• Figure 2: The set $Q^{(1)}$ and the eigenvalue $\Omega$ with branch points $\lambda_1=2\mathrm{i}/5$, $\lambda_2=4\mathrm{i} /5$, $\lambda_3=7\mathrm{i}/5$.
• Figure 3: The set $Q$ and the related eigenvalues $\Omega(\lambda)$ under the \ref{['case2']}. The related numerical spectrums are consistent with the recent work CuiP-2025.
• Figure 4: The maximum value of the parameter $P$ under the modulus $k_2^{(2)}$.
• Figure 5: The correspondence between the $\nu$-plane and $\lambda$-plane with branch points satisfying \ref{['case1']}. From top to bottom, the green, blue, yellow, green, red, yellow points in the $\lambda$-plane correspond to the branch points $\lambda_3$, $\lambda_2$, $\lambda_1$, $\lambda_1^*$, $\lambda_2^*$ and $\lambda_3^*$, respectively.

Theorems & Definitions (74)

• Proposition 1
• Definition 1: Riemann theta function BelokolosBME-1986
• Definition 2
• Theorem 1
• Proposition 2
• Theorem 2
• Theorem 2
• Definition 3
• Theorem 3
• Theorem 4