Integrability structures of the $(2+1)$-dimensional Euler equation

TL;DR

This paper explores the integrability of the (2+1)-dimensional Euler equation by constructing Hamiltonian and symplectic structures, analyzing their effects on symmetries and cosymmetries.

Contribution

It introduces new local and nonlocal Hamiltonian and variational symplectic structures for the (2+1)-dimensional Euler equation, advancing understanding of its integrability.

Findings

Constructed local and nonlocal Hamiltonian structures

Analyzed the action on cosymmetries and contact symmetries

Enhanced understanding of the equation's integrability properties

Abstract

We construct local and nonlocal Hamiltonian structures and variational symplectic structures for the $(2+1)$-dimensional Euler equation in the vorticity form and study the action of the local Hamiltonian and symplectic structures on the cosymmetries of second order and the contact symmetries.