The N -soliton solution of the Degasperis–Procesi equation

This paper extends the results of a previous paper designated I hereafter in which the one- and two-soliton solutions of the Degasperis–Procesi (DP) equation were obtained and their peakon limit was considered. Here, we present the general N-soliton solution of the DP equation and investigate its property. We show that it has a novel structure expressed by a parametric representation in terms of the BKP τ-functions. A purely algebraic proof of the solution is given by establishing various identities among the τ-functions. The large time asymptotic of the solution recovers the formula for the phase shift which was derived in I by a different method. Finally, the structure of the N-soliton solution is discussed in comparison with that of the Camassa–Holm shallow water wave equation.