Dispersionless integrable systems in 3D and Einstein-Weyl geometry. (English) Zbl 1306.37084

The authors consider four classes of dispersionless partial differential equations in the variables \((x,y,t) \in \mathbb{R}^3\) which appear in mathematical physics, general relativity, differential geometry and the theory of integrable systems. Although each of the four classes are treated separately, there are two main directions of study, and results, which apply to all classes (with an extra assumption on the last class). In the formal linearization, the coefficients of the PDEs can be interpreted as symmetric tensors fields. The solutions to the equations have thus an associated conformal structure essentially identified with the class \([g]\) of a Lorentzian metric, hence a \((0,2)\)-tensor, \(g \in \Gamma (S^2T^*M)\), where \(M\) is the base manifold with coordinates \((x,y,t)\). The first main result states that the equations are linearizable if and only if the conformal structure \(g\) is conformally flat on every solution. The second main result shows that the equations are integrable if and only if the conformal structure \(g\) is Einstein-Weyl on every solution. The paper is self-contained with detailed proofs. As the signature of the metric corresponds to the hyperbolicity of the equations, it is worth mentioning that the authors also study the elliptic case in one of the last sections.


37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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