Multisoliton solutions of the Degasperis-Procesi equation and their peakon limit. (English) Zbl 1086.35095

Matsuno, Yoshimasa

Inverse Probl. 21, No. 5, 1553-1570 (2005).

Summary: The multisoliton solutions of the Degasperis-Procesi (DP) equation are constructed by means of a reduction procedure (CKP reduction) for the multisoliton solutions of the Kadomtsev-Petviashvili hierarchy. The solutions have parametric representations and exhibit several new features when compared with existing soliton solutions. Of particular interest are the one- and two-soliton solutions for which a detailed analysis is performed. The explicit formula for the phase shift is obtained which occurs in the interaction process of two solitons. We find that the soliton velocity depends nonlinearly on its amplitude as opposed to the usual linear relation. Also, the interaction of two solitons reveals that the slow soliton exhibits a nonnegative phase shift in a certain range of the wave parameters.
Subsequently, we consider the peakon limit of the soliton solutions and show that it recovers all the features already reported for the peakon solutions of the DP equation. We also derive the asymptotic form of the general \(N\)-soliton solution as well as the formula for the phase shift.

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
37K40	Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q51	Soliton equations

Keywords:

phase shift; interaction; peakon solutions; \(N\)-soliton solution