Multi-variable reductions of the dispersionless DKP hierarchy

TL;DR

This paper studies multi-variable reductions of the dispersionless DKP hierarchy using elliptic parametrization, deriving compatibility conditions, and proving solvability of the resulting hydrodynamic system with an Egorov metric.

Contribution

It introduces elliptic L"owner equations for the dispersionless DKP hierarchy and establishes their compatibility and solvability, extending the theory of integrable systems.

Findings

Derived elliptic L"owner equations for the hierarchy

Proved compatibility conditions as elliptic Gibbons-Tsarev equations

Established solvability via the generalized hodograph method

Abstract

We consider multi-variable reductions of the dispersionless DKP hierarchy (the dispersionless limit of the Pfaff lattice) in the elliptic parametrization. The reduction is given by a system of elliptic L\"owner equations supplemented by a system of partial differential equations of hydrodynamic type. The compatibility conditions for the elliptic L\"owner equations are derived. They are elliptic analogues of the Gibbons-Tsarev equations. We prove solvability of the hydrodynamic type system by means of the generalized hodograph method. The associated diagonal metric is proved to be of the Egorov type.