On Integrability of the Equations for Nonsingular Pairs of Compatible Flat Metrics

TL;DR

This paper develops a method to integrate nonlinear equations describing nonsingular pairs of compatible flat metrics in multiple dimensions, using a reduction to Lame equations and the Zakharov method within the inverse scattering framework.

Contribution

It introduces a novel reduction of the problem to Lame equations and applies the Zakharov method to solve for compatible flat metrics in the general N-component case.

Findings

Provides a scheme for integrating the equations for compatible flat metrics.

Reduces the problem to a special case of Lame equations.

Utilizes the Zakharov method of differential reductions in the dressing method.

Abstract

We deal with the problem of description of nonsingular pairs of compatible flat metrics for the general $N$-component case. We describe the scheme of the integrating the nonlinear equations describing nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics). It is based on the reducing this problem to a special reduction of the Lame equations and the using the Zakharov method of differential reductions in the dressing method (a version of the inverse scattering method).