# Relativistic DNLS and Kaup-Newell Hierarchy

## TL;DR

This paper constructs a relativistic derivative NLS equation using the Kaup-Newell hierarchy's recursion operator, demonstrating its integrability and explicit linear problem representation, and showing it reduces to the DNLS equation in the nonrelativistic limit.

## Contribution

It introduces a relativistic derivative NLS equation derived from the Kaup-Newell hierarchy and provides a compact linear problem representation using q-calculus.

## Key findings

- The relativistic DNLS equation maintains integrability at all relativistic orders.
- The linear problem for the relativistic DNLS is explicitly represented using q-calculus.
- In the nonrelativistic limit, the equation reduces to the standard DNLS equation.

## Abstract

By the recursion operator of the Kaup-Newell hierarchy we construct the relativistic derivative NLS (RDNLS) equation and the corresponding Lax pair. In the nonrelativistic limit $c \rightarrow \infty$ it reduces to DNLS equation and preserves integrability at any order of relativistic corrections. The compact explicit representation of the linear problem for this equation becomes possible due to notions of the $q$-calculus with two bases, one of which is the recursion operator, and another one is the spectral parameter.