Modulational Stability of Wave Trains in the Camassa-Holm Equation

Abstract

In this paper, we study the nonlinear wave modulation of arbitrary amplitude periodic traveling wave solutions of the Camassa-Holm (CH) equation. Slow modulations of wave trains is often described through Whitham's theory of modulations, which at leading order models the slow evolution of the fundamental wave characteristics (such as the wave's frequency, mass and momentum) through a disperionless system of quasi-linear partial differential equations. The modulational stability or instability of such a slowly modulated wave is considered to be determined by the hyperbolicity or ellipticity of this Whitham modulation system of equations. In work by Abenda \& Grava, the Whitham modulation system for the CH equation was derived through averaged Lagrangian methods and was further shown to always be hyperbolic (although strict hyperbolicity may fail). In this work, we provide an independent derivation of the Whitham modulation system for the CH equation through nonlinear WKB / multiple scales expansions. We further provide a rigorous connection between the Whitham modulation equations for the CH equation and the spectral stability of the underlying periodic wave train to localized (i.e. integrable on the line) perturbations. In particular, we prove that the strict hyperbolicity of the Whitham system implies spectral stability in a neighborhood of the origin in the spectral plane, i.e. spectral modulational stability. As an illustration of our theory, we examine the Whitham modulation system for wave trains with asymptotically small oscillations about their total mass.

Figures (2)

• Figure 1: Depiction of the effective potential $V(\phi;a,c)$ for an admissible value of $a\in\mathbb R$. Note that there is a vertical asymptote at $\phi=c$, and that all the periodic solutions here exist for $\phi<c$, oscillating around the critical points $\phi_2$ of the potential $V$.
• Figure 2: A numerically generated plot of the intersection of the existence region $\mathcal{M}$ and the plane $c=2$. The upper and lower black curves correspond to solitary waves and constant solutions, respectively, and act as the boundary of the existence region. Note these upper and lower boundaries intersect at $(a,E) = (4c^3/27,c^2/6)$, while they intersect the left boundary $a=0$ at $E=0$ and $E=-c^2/2$, respectively (recall here $c=2$).

Theorems & Definitions (15)

• Remark 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Lemma 2.1: Period Monotonicity
• Remark 2.2
• Proposition 2.3
• Remark 2.4
• Remark 4.1
• Definition 4.2