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

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-like (Korteweg-de Vries) equations 37K40 Soliton theory, asymptotic behavior of solutions 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 35Q51 Soliton-like equations