# Variational Operators, Symplectic Operators, and the Cohomology of Scalar Evolution Equations

## TL;DR

This paper investigates the cohomology of scalar evolution equations, revealing the structure of variational and symplectic operators, and characterizing third-order equations with specific symplectic properties.

## Contribution

It establishes the isomorphism between cohomology spaces and variational or symplectic operators, and characterizes third-order equations with first-order symplectic operators.

## Key findings

- Cohomology space H^{1,2} is isomorphic to variational and symplectic operators.
- H^{1,s} vanishes for s≥3, simplifying the cohomology structure.
- Characterization of third-order scalar evolution equations with first-order symplectic operators.

## Abstract

For a scalar evolution equation $u_t=K(t,x,u,u_x,\ldots, u_n), n\geq 2$ the cohomology spaces $H^{1,s}({\mathcal R}^\infty)$ vanishes for $s\geq 3$ while the space $H^{1,2}({\mathcal R}^\infty)$ is isomorphic to the space of variational operators. The cohomology space $H^{1,2}({\mathcal R}^\infty)$ is also shown to be isomorphic to the space of symplectic operators for $u_t=K$ for which the equation is Hamiltonian. Third order scalar evolution equations admitting a first order symplectic (or variational) operator are characterized. The symplectic nature of the potential form of a bi-Hamiltonian evolution equation is also presented.