On the tau-functions of the Degasperis-Procesi equation

TL;DR

This paper explores the tau-functions of the Degasperis-Procesi equation using determinant and pfaffian identities, constructing N-soliton solutions via a reciprocal transformation linked to a Toda system.

Contribution

It introduces a novel pfaffian formulation of the N-soliton solutions of the DP equation based on determinant-pfaffian identities and a pseudo 3-reduction of the Toda system.

Findings

N-soliton solutions expressed as pfaffians

Established bilinear equations and tau-functions

Linked DP equation solutions to Toda system reductions

Abstract

The DP equation is investigated from the point of view of determinant-pfaffian identities. The reciprocal link between the Degasperis-Procesi (DP) equation and the pseudo 3-reduction of the $C_{\infty}$ two-dimensional Toda system is used to construct the N-soliton solution of the DP equation. The N-soliton solution of the DP equation is presented in the form of pfaffian through a hodograph (reciprocal) transformation. The bilinear equations, the identities between determinants and pfaffians, and the $\tau$-functions of the DP equation are obtained from the pseudo 3-reduction of the $C_{\infty}$ two-dimensional Toda system.