Abstract
We solve the problem of description of nonsingular pairs of compatible flat metrics for the general N‐component case. The integrable nonlinear partial differential equations describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) are found and integrated. The integrating of these equations is based on reducing to a special nonlinear differential reduction of the Lamé equations and using the Zakharov method of differential reductions in the dressing method (a version of the inverse scattering method).
Highlights
Introduction and basic definitionsWe use both contravariant metrics gij(u) with upper indices, where u = (u1, . . . , uN) are local coordinates, 1 ≤ i, j ≤ N, and covariant metrics gij (u) with lower indices, gis(u)gsj (u) = δji
Definition 1.3 is equivalent to our Definition 1.2 of a pencil of metrics of constant Riemannian curvature or, in other words, a compatible pair of the corresponding nonlocal Poisson structures of hydrodynamic type, which were introduced and studied by the author and Ferapontov in [30]
In the general form, the problem of description of flat pencils of metrics was considered by Dubrovin in [6, 7] in connection with the construction of important examples of such flat pencils of metrics generated by natural pairs of flat metrics on the spaces of orbits of Coxeter groups and on other Frobenius manifolds and associated with the corresponding quasi-homogeneous solutions of the associativity equations
Summary
A pencil of metrics is called nonsingular if it is formed by a nonsingular pair of metrics These definitions are motivated by the theory of compatible Poisson brackets of hydrodynamic type. Relation (1.8) gives the condition that an arbitrary linear combination of the metrics g1ij (u) and g2ij (u), (1.6), is a metric of constant Riemannian curvature λ1K1 + λ2K2 In this case, Definition 1.3 is equivalent to our Definition 1.2 of a pencil of metrics of constant Riemannian curvature or, in other words, a compatible pair of the corresponding nonlocal Poisson structures of hydrodynamic type, which were introduced and studied by the author and Ferapontov in [30]. This paper is devoted to the problem of description of all nonsingular pairs of compatible flat metrics and to integrability of the corresponding nonlinear partial differential equations by the inverse scattering method
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