Dispersionless integrable hierarchies and GL(2) geometry

Abstract: (joint work with Evgeny Ferapontov)

Paraconformal or GL(2) geometry on an n-dimensional manifold M is defined by a field of rational normal curves of degree n - 1 in the projectivized cotangent bundle $\mathbb{P}T^*M$. In dimension n=3 this is nothing but a Lorentzian metric. GL(2) geometry is known to arise on solution spaces of ODEs with vanishing Wünschmann invariants.

We show that GL(2) structures also arise on solutions of dispersionless integrable hierarchies of PDEs such as the dispersionless Kadomtsev-Petviashvili (dKP) hierarchy. In fact, they coincide with the characteristic variety (principal symbol) of the hierarchy. GL(2) structures arising in this way possess the property of involutivity. For n=3 this gives the Einstein-Weyl geometry.

Thus we are dealing with a natural generalization of the Einstein-Weyl geometry. Our main result states that involutive GL(2) structures are governed by a dispersionless integrable system whose general local solution depends on 2n - 4 arbitrary functions of 3 variables. This establishes integrability of the system of Wünschmann conditions.

mathematical physicsanalysis of PDEsdifferential geometry

Audience: researchers in the topic

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Geometry of differential equations seminar