Communications in Analysis and Geometry

Volume 18 (2010)

Number 4

Integrable GL(2) geometry and hydrodynamic partial differential equations

Pages: 743 – 790



Abraham D. Smith (Department of Mathematics and Statistics, McGill University, Montreal, Canada)


This article is a local analysis of integrable ${\rm GL}(2)$-structures of degree 4.A ${\rm GL}(2)$-structure of degree $n$ corresponds to a distribution of rationalnormal cones over a manifold of dimension $n+1$. Integrability corresponds tothe existence of many submanifolds that are spanned by lines in the cones.These ${\rm GL}(2)$-structures are important because they naturally arise from acertain family of second-order hyperbolic partial differential equations (PDEs) in three variables that areintegrable via hydrodynamic reduction. Familiar examples include the waveequation, the first flow of the dKP hierarchy and the Boyer–Finley equation.

The main results are a structure theorem for integrable ${\rm GL}(2)$-structures, aclassification for connected integrable ${\rm GL}(2)$-structures and an equivalencebetween local integrable ${\rm GL}(2)$-structures and Hessian hydrodynamic hyperbolicPDEs in three variables.

This yields natural geometric characterizations of the wave equation, thefirst flow of the dKP hierarchy and several others. It also provides anintrinsic, coordinate-free infrastructure to describe a large class of hydrodynamicintegrable systems in three variables.

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