Lagrangian and Hamiltonian Structures for the Constant Astigmatism Equation

TL;DR

This paper develops a Lagrangian and Hamiltonian framework for the constant astigmatism equation, enabling the construction of integrable flows and conservation laws through bi-Hamiltonian structures.

Contribution

It introduces a novel Lagrangian and Hamiltonian formulation for the constant astigmatism equation, including a bi-Hamiltonian structure and an infinite hierarchy of commuting flows.

Findings

Established a Lagrangian representation for the equation.

Constructed a first evolution commuting flow of third order.

Presented a second Hamiltonian structure and bi-Hamiltonian hierarchy.

Abstract

In this paper we found a Lagrangian representation and corresponding Hamiltonian structure for the constant astigmatism equation. Utilizing this Hamiltonian structure and extra conservation law densities we construct a first evolution commuting flow of the third order. Also, we apply the recursion operator and present a second Hamiltonian structure. This bi-Hamiltonian structure allows to replicate infinitely many local commuting flows and corresponding local conservation law densities.