# Completion of the Ablowitz-Kaup-Newell-Segur integrable coupling

## TL;DR

This paper introduces a novel completion process for integrable couplings, applying it to the Ablowitz-Kaup-Newell-Segur system, resulting in a hierarchy with bi-Hamiltonian structures, advancing the theory of integrable systems.

## Contribution

It proposes a new method to generate integrable systems via perturbation of spectral problems, specifically completing the Ablowitz-Kaup-Newell-Segur integrable coupling.

## Key findings

- Developed a completion process for integrable couplings.
- Constructed a hierarchy with bi-Hamiltonian structures.
- Applied the method to the AKNS integrable coupling.

## Abstract

Integrable couplings are associated with non-semisimple Lie algebras. In this paper, we propose a new method to generate new integrable systems through making perturbation in matrix spectral problems for integrable couplings, which is called the `completion process of integrable couplings'. As an example, the idea of construction is applied to the Ablowitz-Kaup-Newell-Segur integrable coupling. Each equation in the resulting hierarchy has a bi-Hamiltonian structure furnished by the component-trace identity.