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arXiv · 2111.09759

The quadric ansatz for the $mn$-dispersionless KP equation, and supersymmetric Einstein-Weyl spaces

Maciej Dunajski·Prim Plansangkate
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Abstract

We consider two multi-dimensional generalisations of the dispersionless Kadomtsev-Petviashvili (dKP) equation, both allowing for arbitrary dimensionality, and non-linearity. For one of these generalisations, we characterise all solutions which are constant on a central quadric. The quadric ansatz leads to a second order ODE which is equivalent to Painleve I or II for the dKP equation, but fails to pass the Painlev\'e test in higher dimensions. The second generalisation of the dKP equation leads to a class of Einstein-Weyl structures in an arbitrary dimension, which is characterised by the existence of a weighted parallel vector field, together with further holonomy reduction. We construct and characterise an explicit new family of Einstein-Weyl spaces belonging to this class, and depending on one arbitrary function of one variable.

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Maciej Dunajski, Prim Plansangkate. 2021-11-18. The quadric ansatz for the $mn$-dispersionless KP equation, and supersymmetric Einstein-Weyl spaces. https://doi.org/10.1088/1751-8121%2Fac5160

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