On some linear equations associated with dispersionless integrable systems

TL;DR

This paper introduces a matrix extension scheme for dispersionless integrable systems, leading to linear equations linked to Abelian gauge fields and solutions expressed via Lax pair wave functions, enhancing understanding of their linearization.

Contribution

It presents a novel matrix extension approach for dispersionless integrable systems, connecting linear equations with Abelian gauge fields and providing explicit solution constructions.

Findings

Linear equations associated with dispersionless systems are derived.

Solutions are expressed through wave functions of the Lax pair.

The approach offers a new perspective on linearization of integrable systems.

Abstract

We use a recently proposed scheme of matrix extension of dispersionless integrable systems for the Abelian case, in which it leads to linear equations, connected with the initial dispersionless system. In the examples considered, these equations can be interpreted in terms of Abelian gauge fields on the geometric background defined by the dispersionless system. They are also connected with the linearisation of initial systems. We construct solutions to these linear equations in terms of wave functions of the Lax pair for dispersionless system, which is represented in terms of some vector fields.