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Integrable \(\mathrm{GL}(2)\) geometry and hydrodynamic partial differential equations. (English) Zbl 1239.58020

From the author’s abstract: “This article is a local analysis of integrable \(\mathrm{GL}(2)\)-structures of degree 4. A \(\mathrm{GL}(2)\)-structure of degree n corresponds to a distribution of rational normal cones over a manifold \(M\) of dimension \((n+1)\). Integrability corresponds to the existence of many submanifolds that are spanned by lines in the cones. These \(\mathrm{GL}(2)\)-structures are important because they naturally arise from a certain family of second-order hyperbolic PDEs in three variables that are integrable via hydrodynamic reduction. Familiar examples include the wave equation, the first flow of the dKP equation, and the Boyer-Finley equation. The main results are a structure theorem for integrable \(\mathrm{GL}(2)\)-structures, a classification for connected integrable \(\mathrm{GL}(2)\)-structures, and an equivalence between local integrable \(\mathrm{GL}(2)\)-structures and Hessian hydrodynamic hyperbolic PDEs in three variables. This yields natural geometric characterizations of the wave equation, the first flow of the dKP hierarchy, and several others. It also provides an intrinsic, coordinate-free infrastructure to describe a large class of hydrodynamic integrable systems in three variables.”

MSC:

58J45 Hyperbolic equations on manifolds
58A15 Exterior differential systems (Cartan theory)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q35 PDEs in connection with fluid mechanics
53C10 \(G\)-structures
58A20 Jets in global analysis
35A30 Geometric theory, characteristics, transformations in context of PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L05 Wave equation
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