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Abstract

The nonlinear partial differential equationarises as a model for the unidirectional propagation of shallow water waves over a flat bottom, with u(x, t) representing the water's free surface in nondimensional variables. In this paper, we are concerned with the periodic solutions of (1.1), that is, u : S × [0, T ) → R where S denotes the unit circle and T > 0 is the maximal existence time of the solution. Equation (1.1) was first obtained [12] as an abstract bi-Hamiltonian equation with infinitely many conservation laws and was subsequently derived from physical principles [2]. For a discussion of the physical relevance of (1.1) in the context of water waves, we refer to [14, 15]. Equation (1.1) also arises as a model for nonlinear waves in cylindrical axially symmetric hyperelastic rods, with u(x, t) representing the radial stretch relative to a prestressed state [11]. Moreover, (1.1) is a re-expression of the geodesic flow in the group of compressible diffeomorphisms of the circle [18], just like the Euler equation is an expression of the geodesic flow in the group of incompressible diffeomorphisms of the torus [1]. This geometric interpretation leads to a proof that equation (1.1) satisfies the least action principle [7]: a state of the system is transformed to another nearby state through a uniquely determined flow that minimizes the energy. We also point out that for a large class of initial data, equation (1.1) is an infinite-dimensional completely integrable Hamiltonian system: by means of an isospectral problem, one can convert the equation into an infinite sequence of linear ordinary differential equations which can be

Orbital stability of periodic peakons for the generalized modified Camassa-Holm equation

“…To prove the above main result, we recall an intriguing works for the stability of peakons for the CH equation (1.1) [15,16,20,35,36] and mCH equation (1.3) [40,52]. The CH equation (1.1) and the mCH equation (1.3) has the following useful conservation laws:…”

Orbital stability of elliptic periodic peakons for the modified Camassa-Holm equation

“…and ϕ(x) is defined for x ∈ [0, 1) and extends periodically to the whole real line. Orbital stability of the periodic peakons for the CH equation was proved by Lenells in [20,21]. Wang and Tian [29] extended Lenell's approach to prove the orbital stability of the periodic peakons for the Novikov equation.…”

“…The proof is inspired by [21] where the case of hyperbolic periodic peakons of (1.4) is considered. The approach taken here is similar but there are differences.…”

“…As Lenells said in [21], for a wave profile u ∈ H 1 (S), the functional H 0 [u] and H 1 [u] represents the momentum and kinetic energy, respectively. Lemma 3.1 says that if a wave u ∈ H 1 (S) has momentum H 0 [u], energy H 1 [u] and height M u close to the elliptic periodic peakon's momentum, energy and height, then the whole shape of u is close to that of the elliptic periodic peakon.…”

Orbital Stability of Peakons for a Generalized Camassa-Holm Equation

“…The approach in [13] was extended to prove the orbital stability of the peakons for the other nonlinear wave equations [14]- [25]. The method of proved orbital stable of peakons was also extended to periodic peakons [26] [27] [28] [29] [30]. In [11], Anco and Recio obtained an interesting generalization of the Camassa-Holm and FORQ/modified Camassa-Holm equations by deriving the most general subfamily of peakon equations that possess the Hamiltonian structure shared by the Camassa-Holm and FORQ/modified Camassa-Holm equations.…”