Dispersionless BKP equation, the Manakov-Santini system and Einstein-Weyl structures

TL;DR

This paper establishes a connection between solutions of the dispersionless BKP equation and the Manakov-Santini system, leading to the construction of Einstein-Weyl structures and extending to the BMS system through a Miura transformation.

Contribution

It introduces a novel mapping from the dispersionless BKP equation to the Manakov-Santini system and extends this to the BMS system, revealing new geometric structures.

Findings

Mapped solutions of dBKP to MS system

Constructed Einstein-Weyl structures for these solutions

Extended the map to BMS system via Miura transformation

Abstract

We construct a map of solutions of the dispersionless BKP (dBKP) equation to solutions of the Manakov-Santini (MS) system. This map defines an Einstein-Weyl structure corresponding to the dBKP equation through the general Lorentzian Einstein-Weyl structure corresponding to the MS system. We give a spectral characterisation of reduction of the MS system which singles out the image of the dBKP equation solutions and also consider more general reductions of this class. We define the BMS system and extend the map defined above to the map (Miura transformation) of solutions of the BMS system to solutions of the MS system, thus obtaining an Einstein-Weyl structure for the BMS system.