Ghostpeakons and Characteristic Curves for the Camassa-Holm, Degasperis-Procesi and Novikov Equations

Abstract

We derive explicit formulas for the characteristic curves associated with the multipeakon solutions of the Camassa-Holm, Degasperis-Procesi and Novikov equations. Such a curve traces the path of a fluid particle whose instantaneous velocity equals the elevation of the wave at that point (or the square of the elevation, in the Novikov case). The peakons themselves follow characteristic curves, and the remaining characteristic curves can be viewed as paths of "ghostpeakons" with zero amplitude; hence, they can be obtained as solutions of the ODEs governing the dynamics of multipeakon solutions. The previously known solution formulas for multipeakons only cover the case when all amplitudes are nonzero, since they are based upon inverse spectral methods unable to detect the ghostpeakons. We show how to overcome this problem by taking a suitable limit in terms of spectral data, in order to force a selected peakon amplitude to become zero. Moreover, we use direct integration to compute the characteristic curves for the solution of the Degasperis-Procesi equation where a shockpeakon forms at a peakon-antipeakon collision. In addition to the theoretical interest in knowing the characteristic curves, they are also useful for plotting multipeakon solutions, as we illustrate in several examples.

Instability of H1-stable peakons in the Camassa–Holm equation

“…Multi-peakon solutions were constructed by many analytical and numerical tools [1,6,18,25]. The local characteristic curve x = ξ(t) for the Camassa-Holm equation (1.2) is defined by the equation…”

Non-interlacing peakon solutions of the Geng–Xue equation

“…However, the equation for x k (t ) in the peakon ODEs is still nontrivial and its solution describes the trajectory of a zero-amplitude ghostpeakon which is influenced by the other peakons but does not influence them. These ghostpeakon trajectories are of interest, since they are in fact the characteristic curves associated with the multipeakon solution, and we recently derived explicit formulas for the ghostpeakons for the CH, DP and Novikov equations [10]. Moreover, as we will demonstrate in this paper, the methods used for studying ghostpeakons are also very useful for studying ordinary (non-ghost) peakons for the GX equation.…”

“…(6.1) Remark 6.2. The inspiration for this method comes from our previous paper [10], where we used ghostpeakons to compute the characteristic curves for peakon solutions of the Camassa-Holm, Degasperis-Procesi and Novikov equations.…”

“…clear that it is going to work, since the Lax pairs for the Geng-Xue equation do not seem to provide sufficiently many constants of motion in the non-interlacing case; see Section 11. Motivated by our previous paper [10], we will instead use the idea of ghostpeakons: starting from an interlacing peakon solution with a larger number of peakons, we perform appropriate limiting procedures on the spectral data to "kill off" selected peakons, i.e., we turn them into ghostpeakons by making their amplitudes tend to zero. After discarding the ghostpeakons and relabeling the variables, what remains is the desired peakon configuration.…”

Peakons of Novikov Equation via the Homotopy Analysis Method

“…In [5], explicit multipeakon solutions of NE were calculated by inverse spectral methods using the matrix Lax pair found in [7]. In [17], explicit formulas were given for the characteristic curves of the CH, DP, and Novikov equations. In particular, Section 2 of [17] consists of a very good survey on the existing knowledge on these three equations.…”