# Linear integral equations, infinite matrices, and soliton hierarchies

## TL;DR

This paper introduces a systematic framework connecting linear integral equations and infinite matrices to construct and analyze hierarchies of soliton equations, including their integrability features and solutions.

## Contribution

It provides a unified approach to derive soliton hierarchies and their properties from scalar linear integral equations and infinite matrix representations.

## Key findings

- Constructed hierarchies of soliton equations in (2+1) and (1+1) dimensions.
- Derived integrability features such as Lax pairs, $ au$-functions, and soliton solutions.
- Established connections between integral equations, infinite matrices, and soliton hierarchies.

## Abstract

A systematic framework is presented for the construction of hierarchies of soliton equations. This is realised by considering scalar linear integral equations and their representations in terms of infinite matrices, which give rise to all (2+1)- and (1+1)-dimensional soliton hierarchies associated with scalar differential spectral problems. The integrability characteristics for the obtained soliton hierarchies, including Miura-type transforms, $\tau$-functions, Lax pairs as well as soliton solutions, are also derived within this framework.