• DOI:10.1090/conm/593/11873
• Corpus ID: 118231987

Peakon-antipeakon interactions in the Degasperis-Procesi Equation

@article{Szmigielski2013PeakonantipeakonII,
  title={Peakon-antipeakon interactions in the Degasperis-Procesi Equation},
  author={Jacek Szmigielski and Lingjun Zhou},
  journal={arXiv: Mathematical Physics},
  year={2013}
}

• J. Szmigielski, Lingjun Zhou
• Published 2 January 2013
• Mathematics, Physics
• arXiv: Mathematical Physics

Peakons are singular, soliton-like solutions to nonlinear wave equations whose dynamics can be studied using ordinary differential equations (ODEs). The Degasperis-Procesi equation (DP) is an important example of an integrable PDE exhibiting wave breaking in the peakon sector thus affording an interpretation of wave breaking as a mechanical collision of particles. In this paper we set up a general formalism in which to study collisions of DP peakons and apply it, as an illustration, to a detailed study of three colliding peakons. It is shown that peakons can collide only in pairs, no triple collisions are allowed and at the collision a shockpeakon is created. We also show that the initial configuration of peakon-antipeakon pairs is nontrivially correlated with the spectral properties of an accompanying non-selfadjoint boundary value problem. In particular if peakons or antipeakons are bunched up on one side relative to a remaining antipeakon or peakon then the spectrum is real and simple. Even though the spectrum is in general complex the existence of a global solution in either time direction dynamics is shown to imply the reality of the spectrum of the boundary value problem.