Dispersionless BKP Equation, the Manakov–Santini System and Einstein–Weyl Structures

Abstract

We construct a map from 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 in the MS system, which singles out the image of the dBKP equation solution, 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.

Special Issue Editorial “Symmetry of Hamiltonian Systems: Classical and Quantum Aspects”

“…Another very interesting and geometrically motivated trend is presented by the article by L.V. Bogdanov [3], devoted to studying a map from solutions of the dispersionless BKP (dBKP) equation to solutions of the Manakov-Santini (MS) system. It is shown that this map defines an Einstein-Weyl structure corresponding to the dBKP equation through the general Lorentzian Einstein-Weyl structure, corresponding to the MS system.…”