# A Variational Approach to Lax Representations

## TL;DR

This paper demonstrates that the Zakharov-Mihailov Lagrangian structure for integrable equations has a multiform structure, enabling a variational principle for the Lax pair, with examples including the AKNS hierarchy.

## Contribution

It introduces a Lagrangian multiform framework for Lax representations, allowing variational principles directly on the Lax pair, a novel approach in integrable systems.

## Key findings

- Established Lagrangian multiform structure for ZM Lagrangian
- Formulated variational principle for Lax pairs
- Applied framework to AKNS hierarchy examples

## Abstract

It is shown that the Zakharov-Mihailov (ZM) Lagrangian structure for integrable nonlinear equations derived from a general class of Lax pairs possesses a Lagrangian multiform structure. We show that, as a consequence of this multiform structure, we can formulate a variational principle for the Lax pair itself, a problem that to our knowledge was never previously considered. As an example, we present an integrable $N\times N$ matrix system that contains the AKNS hierarchy, and we exhibit the Lagrangian multiform structure of the scalar AKNS hierarchy by presenting the components corresponding to the first three flows of the hierarchy.