Lagrangian Approach to Dispersionless KdV Hierarchy

TL;DR

This paper develops a Lagrangian framework to analyze the Hamiltonian structure of dispersionless KdV hierarchies, enhancing the understanding of symmetries and conserved quantities in integrable systems.

Contribution

It introduces a Lagrangian approach that complements existing r-matrix methods for dispersionless KdV hierarchies, providing new tools for constructing conserved densities and Hamiltonian operators.

Findings

Lagrangian formulation clarifies symmetry-conservation relations

New methods for deriving conserved densities and Hamiltonians

Enhanced understanding of the variational symmetries in dispersionless KdV

Abstract

We derive a Lagrangian based approach to study the compatible Hamiltonian structure of the dispersionless KdV and supersymmetric KdV hierarchies and claim that our treatment of the problem serves as a very useful supplement of the so-called r-matrix method. We suggest specific ways to construct results for conserved densities and Hamiltonian operators. The Lagrangian formulation, via Noether's theorem, provides a method to make the relation between symmetries and conserved quantities more precise. We have exploited this fact to study the variational symmetries of the dispersionless KdV equation.