Boyer Finley equation and systems of hydrodynamic type

Abstract

We reduce Boyer-Finley equation to a family of compatible systems of hydrodynamic type, with characteristic speeds expressed in terms of spaces of rational functions. The systems of hydrodynamic type are then solved by the generalized hodograph method, providing solutions of the Boyer-Finley equation including functional parameters. In this paper we construct solutions of the dispersionless non-linear PDE -the Boyer-Finley equation (self-dual Einstein equation with a Killing vector),via reduction to a family of compatible systems of hydrodynamic type. This equation was actively studied during last twenty years by many authors; we just mention works [1,2,3,4,5,6,7,8,9,14,15]. So far the most general scheme of the construction of its solutions was developed in [8,9]. In these works solutions of the Boyer-Finley equation were derived by averaging an appropriate two-point Baker-Akhiezer function in genus zero which corresponds to

On the Integrability of (2+1)-Dimensional Quasilinear Systems

“…For any solution µ i , v of the system (16) one can reconstruct λ i and w by virtue of (13). In the twocomponent case this system takes the form…”

Integrable Hydrodynamic Chains

“…It is easy to check, that all possible hydrodynamic reductions of this system (61) (see, for instance, approach in [25])…”

“…For instance, this problem for the Benney momentum chain is still open (see [19]). However, this problem for the Boyer-Finley momentum chain (continuum limit of the DarbouxLaplace chain, which also is known as two-dimensional Toda lattice, see [3], [25] and [22]) in fact is not exist, because both mentioned hydrodynamic chains are related by special exchange of independent variables, see [21]). Thus, we are lucky to solve this problem for the hydrodynamic chain (37).…”

The Haantjes tensor and double waves for multi-dimensional systems of hydrodynamic type: a necessary condition for integrability

“…These solutions were extensively investigated in gas dynamics and magnetohydrodynamics in a series of publications [39,2,3,4,34,35,7,21]. Later, they reappeared in the context of the dispersionless KP and Toda hierarchies [17,18,19,22,28,29,42,10], the theory of integrable hydrodynamic chains [31,32,30] and the Laplacian growth problems [26]. In [13], it was suggested to call a multi-dimensional system integrable if, for arbitrary n, it possesses infinitely many n-component reductions of the form (9) parametrized by n arbitrary functions of a single argument.…”