An Application of Pfaffians to multipeakons of the Novikov equation and the finite Toda lattice of BKP type

TL;DR

This paper introduces a novel Pfaffian-based approach to analyze multipeakon solutions of the Novikov equation, establishing a connection with the finite B-Toda lattice of BKP type and proposing generalizations.

Contribution

It is the first to interpret the peakon problem using Pfaffians and links Novikov peakons to the finite B-Toda lattice of BKP type.

Findings

Novikov peakon ODEs describe an isospectral flow on a Pfaffian-defined manifold

Established a connection between Novikov peakons and the finite B-Toda lattice of BKP type

Proposed generalizations of the Novikov equation and B-Toda lattice with explicit solutions

Abstract

The Novikov equation is an integrable analogue of the Camassa-Holm equation with a cubic (rather than quadratic) nonlinear term. Both these equations support a special family of weak solutions called multipeakon solutions. In this paper, an approach involving Pfaffians is applied to study multipeakons of the Novikov equation. First, we show that the Novikov peakon ODEs describe an isospectral flow on the manifold cut out by certain Pfaffian identities. Then, a link between the Novikov peakons and the finite Toda lattice of BKP type (B-Toda lattice) is established based on the use of Pfaffians. Finally, certain generalizations of the Novikov equation and the finite B-Toda lattice are proposed together with special solutions. To our knowledge, it is the first time that the peakon problem is interpreted in terms of Pfaffians.