# Superposition of the Coupled NLS and MKdV Systems

## TL;DR

This paper discusses the integrability and solution properties of superpositions of coupled nonlinear Schrödinger (NLS) and modified Korteweg-de Vries (MKdV) systems, emphasizing their relation to existing hierarchies and solutions.

## Contribution

It analyzes the superposition of integrable hierarchies like NLS and MKdV, showing they are not new equations but share solutions and dispersion relations with known integrable systems.

## Key findings

- Superpositions of integrable hierarchies remain integrable.
- Superposed equations do not introduce new solutions, only modified dispersion relations.
- The AKNS system exemplifies these properties in (1+1)-dimensions.

## Abstract

Superpositions of hierarchies of integrable equations are also integrable. The superposed equations, such as the Hirota equations in the AKNS hierarchy, cannot be considered as new integrable equations. Furthermore if one applies the Hirota bilinear method to these equations one obtains the same $N$-soliton solutions of the generating equation which differ only by the dispersion relations. Similar discussions can be made for the locally and nonlocally reduced equations as well. We give, as an example, AKNS system of equations in $(1+1)$-dimensions.