Colliding peakons and the formation of shocks in the Degasperis–Procesi equation

Abstract

The Degasperis-Procesi equation (DP) is one of several equations known to model important nonlinear effects such as wave breaking and shock creation. It is, however, a special property of the DP equation that these two effects can be studied in an explicit manner with the help of the multi-peakon ansatz. In essence, this ansatz allows one to model wave breaking as a collision of hypothetical particles (peakons and antipeakons), called henceforth collectively multipeakons. It is shown that the system of ordinary differential equations describing DP multi-peakons has the Painlevé property which implies a universal wave breaking behaviour, that multipeakons can collide only in pairs, and that there are no multiple collisions other than, possibly simultaneous, collisions of peakon-antipeakon pairs at different locations. Moreover, it is demonstrated that each peakonantipeakon collision results in creation of a shock thus making possible a multi-shock phenomenon.

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

“…If the eigenvalues are complex and simple, and λ i + λ j = 0 for all i and j, then the formulas work without modification; the eigenvalues occur in complex-conjugate pairs, and whenever λ j = λ i , then also b j = b i , which will make all U a k real-valued, and the formulas provide a solution of the initial value problem until the first collision. If there are eigenvalues of multiplicity greater than one, then the formulas must be modified, taking into account that the partial fraction decomposition of the Weyl function will involve coefficients b (i) k whose time dependence is given by a polynomial in t times the exponential e t/λ k ; see Szmigielski and Zhou [65,66]. There is also the possibility of resonant cases, where λ i + λ j = 0 for one or more pairs (i, j).…”

Lax Integrability and the Peakon Problem for the Modified Camassa–Holm Equation

“…In the following years the peakon ansatz was successfully applied to another, well studied by now, equation, namely the Degasperis-Procesi equation [14] m t + um x + 3u x m = 0, m = u − u xx , (1.4) which despite its superficial similarity to the CH equation (1.2) has in addition shock solutions [9,10,40], while its peakon sector leads to new questions regarding Nikishin systems [49] studied in approximation theory [4,41]. For potential applicability to water wave theory the reader is invited to consult [12]; for a discussion of weak solutions see [18]; [37,39] present important results regarding stability, and finally [58,59] deal with collisions of peakons and the onset of shocks in the form of shockpeakons [40].…”

Multipeakons of a two-component modified Camassa–Holm equation and the relation with the finite Kac–van Moerbeke lattice

“…Not surprisingly, the initial success of CH also prompted researchers to look for other equations exhibiting similar properties, resulting in several new equations that have been proposed and studied over last twenty years. Here, perhaps the most intriguing is the Degasperis-Procesi equation (DP) [14,11] m t + m x u + 3mu x = 0, m = u − u xx , (1.2) which not only supports peakon solutions [30] and wave-breaking [15], but also shocks [29,42]. Other equations that attracted considerable attention are (1) the Novikov equation [23,22,36]…”