Multi-peakon solutions of the Degasperis–Procesi equation
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Abstract
Updates: Corrected an error in the (unnumbered) expression for m3(t) at the end of the paper. Page numbers for Ref. [3] corrected. Ref. [8] added. Abstract. We present an inverse scattering approach for computing n-peakon solutions of the Degasperis–Procesi equation (a modification of the Camassa– Holm (CH) shallow water equation). The associated non-self-adjoint spectral problem is shown to be amenable to analysis using the isospectral deformations induced from the n-peakon solution, and the inverse problem is solved by a method generalizing the continued fraction solution of the peakon sector of the CH equation. Degasperis and Procesi [2] showed, using the method of asymptotic integrability, that the PDE ut − uxxt + (b + 1)uux = buxuxx + uuxxx (1) cannot be completely integrable unless b = 2 or b = 3. The case b = 2 is the Camassa– Holm (CH) shallow water equation [1], which is well known to be integrable and to possess multi-soliton (weak) solutions with peaks, so called multi-peakons. Degasperis,
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