The Stability of the $b$-family of Peakon Equations

Abstract

In the present work we revisit the b-family model of peakon equations, containing as special cases the b = 2 (Camassa-Holm) and b = 3 (Degasperis-Procesi) integrable examples. We establish information about the point spectrum of the peakon solutions and notably find that for suitably smooth perturbations there exists point spectrum in the right half plane rendering the peakons unstable for b < 1. We explore numerically these ideas in the realm of fixed-point iterations, spectral stability analysis and time-stepping of the model for the different parameter regimes. In particular, we identify exact, stationary (spectrally stable) lefton solutions for b < −1, and for −1 < b < 1, we dynamically identify ramp-cliff solutions as dominant states in this regime. We complement our analysis by examining the breakup of smooth initial data into stable peakons for b > 1. While many of the above dynamical features had been explored in earlier studies, in the present work, we supplement them, wherever possible, with spectral stability computations.

Similarity reductions of peakon equations: the $b$-family

“…[43]), which says that for any suitable dispersive evolutionary PDE (not necessarily integrable), generic initial data should decompose into a train of solitary waves together with radiation which decays to zero as t → ∞. Apart from intensive studies of the integrable cases b = 2, 3, further analytical and numerical support for the behaviour reported by Holm and Staley has taken a while to materialize: an orbital stability result for a single lefton when b < −1 was proved in [27], while there are various well-posedness and rigidity results (see [29,38] and references); yet linear stability/instability results for peakons, ramp/cliff solutions and leftons across the full range of b values have been found only very recently [7,31].…”

Existence and regularity for global solutions including breaking waves from Camassa-Holm and Novikov equations to $λ$-family equations

“…when b = λ + 2. Especially, when λ = 0, the equation (1.4) is sometimes called as the b-family equations or b-equations [1,10], including the Degasperis-Procesi equation when b = 3. As a representative equation in the form of (1.1), the Camassa-Holm equation (1.2) was originally derived as a model for surface waves, and has been studied extensively in the past three decades because solutions of this equation enjoy many interesting properties: infinite conservation laws and complete integrability [8,14], existence of peaked solitons and multipeakons [8,9], break down of classical solutions [19,13,18], and formation of finite time breaking waves.…”