On the Lagrangian structure of integrable hierarchies

TL;DR

This paper introduces a continuous pluri-Lagrangian framework for integrable hierarchies, extending lattice system theories to continuous systems and deriving multi-time Euler-Lagrange equations for 2D hierarchies.

Contribution

It develops the concept of pluri-Lagrangian structures for continuous integrable hierarchies, including a formulation for the potential KdV hierarchy, advancing the theoretical understanding.

Findings

Derived multi-time Euler-Lagrange equations for hierarchies

Constructed pluri-Lagrangian formulation for potential KdV hierarchy

Extended lattice theory concepts to continuous integrable systems

Abstract

We develop the concept of pluri-Lagrangian structures for integrable hierarchies. This is a continuous counterpart of the pluri-Lagrangian (or Lagrangian multiform) theory of integrable lattice systems. We derive the multi-time Euler Lagrange equations in their full generality for hierarchies of two-dimensional systems, and construct a pluri-Lagrangian formulation of the potential Korteweg-de Vries hierarchy.