Solitary waves for a higher order Boussinesq system: Stability and numerical experiments

Technical Summary: Solitary Waves for a Higher Order Boussinesq System: Stability and Numerical Experiments

Problem Statement
This work investigates the nonlinear orbital stability of solitary-wave solutions (finite-energy traveling waves) for a class of one-dimensional higher-order Boussinesq evolution systems. The system, denoted as (1.1), models the motion of short waves of small amplitude on the surface of an ideal fluid under gravity, extending the classical first-order $abcd$-Boussinesq systems to a second-order approximation that includes fifth-order dispersive terms. The system possesses a Hamiltonian structure and involves a general nonlinearity $G = (G_1, G_2)^T$ derived from a variational functional $F$. The primary objective is to establish conditions under which the set of ground-state solitary waves is orbitally stable, particularly in the regime where the wave velocity $\omega$ approaches the limiting value of 1.

Methodology
The authors employ a combination of variational methods, asymptotic analysis, and numerical simulations:

1. Variational Framework: The study utilizes the Grillakis–Shatah–Strauss (GSS) theory for orbital stability. Solitary waves are characterized as critical points of an action functional $J_\omega = H + \omega Q$, where $H$ is the Hamiltonian and $Q$ is the charge. The stability of the ground state set $G_\omega$ (minimizers of $J_\omega$) is linked to the convexity of the scalar function $S(\omega) = \inf \{J_\omega(Y) : Y \in M_\omega\}$, specifically the sign of its second derivative $S''(\omega)$.
2. Asymptotic Connection (KdV Scaling): A central technique involves analyzing the behavior of the system as the wave velocity approaches the limit $|\omega| \to 1^-$. By introducing a specific scaling parameter $\epsilon$ such that $\omega^2 = 1 - \epsilon^{4/(p+1)}$, the authors demonstrate that the higher-order Boussinesq system asymptotically converges to a generalized fifth-order Korteweg-de Vries (KdV) equation. This asymptotic reduction allows them to leverage known stability results for the fifth-order KdV equation to infer properties of the Boussinesq system near the limiting velocity.
3. Concentration-Compactness: To prove the existence and compactness of minimizing sequences for the action functional, the authors apply Lions' Concentration-Compactness Principle. They show that vanishing and dichotomy scenarios can be excluded, ensuring that minimizing sequences converge strongly (up to translations) to ground states.
4. Numerical Experiments: The authors develop a Fourier-based iterative scheme to compute solitary-wave profiles and the stability functional $S''(\omega)$. This method adapts techniques from previous work on scalar fifth-order KdV equations, incorporating stabilizing factors to ensure convergence for the coupled Boussinesq system.

Key Contributions and Results

• Existence and Stability via Variational Methods: The paper establishes the existence of ground-state solitary waves for a broad class of homogeneous and non-homogeneous nonlinearities satisfying specific scaling assumptions.
• Stability Criteria: The main theoretical result (Theorem 1.1) provides orbital stability for ground states in the regime $0 < \omega < 1$ under the condition $a + c + 2b = 0$. Specifically:
  • For homogeneous nonlinearities of order $p+2$, stability is guaranteed for $\omega$ sufficiently close to 1 if $0 < p < 8$.
  • For non-homogeneous cases involving mixed homogeneity degrees, stability holds if $p + 2\beta < 8$.
  • The proof relies on showing that $S''(\omega) > 0$ near $\omega = 1$, derived from the asymptotic connection to the generalized fifth-order KdV equation.
• Avoidance of Spectral Analysis: Unlike traditional approaches that require detailed spectral analysis of the linearized operator (which is challenging for this system due to the complexity of the differential operators), this work relies on variational properties and asymptotic estimates, thereby extending stability results to a wider class of nonlinear interactions.
• Numerical Findings: The numerical experiments reveal that the range of wave velocities leading to orbital stability may be larger than the theoretical predictions.
  • For certain parameter sets (e.g., $p=4$), $S''(\omega)$ remains positive across the entire interval $(0, 1)$, suggesting global stability within the physical regime.
  • Conversely, for higher nonlinearity exponents (e.g., $p=9$), $S''(\omega)$ becomes negative near $\omega=1$, indicating instability, which aligns with the theoretical threshold $p < 8$.
  • In non-homogeneous cases (e.g., $G_1 \sim \psi^3, G_2 \sim v^5$), the numerical results show instability in lower velocity ranges while maintaining stability near $\omega = 1$.

Significance and Claims
The authors claim that the primary novelty of this work is the successful application of the GSS framework to higher-order Boussinesq systems without requiring a detailed spectral analysis of the linearized operator. By utilizing the asymptotic connection to the fifth-order KdV equation, they establish orbital stability for a broader class of nonlinearities than previously considered. The paper modestly notes that while the analytical theory proves stability in a neighborhood of $\omega = 1$, the numerical experiments suggest that the stability region might extend further, potentially covering the entire range of physically relevant wave velocities for certain parameter configurations. The work bridges the gap between the theoretical analysis of dispersive systems and numerical verification, offering a robust framework for studying solitary waves in higher-order fluid models.