# Compatible Hamiltonian Operators for the Krichever-Novikov Equation

## TL;DR

This paper demonstrates that the Krichever-Novikov equation's hierarchy is Hamiltonian with a set of compatible operators, including a non-local Hamiltonian operator and two weakly non-local recursion operators, expanding understanding of its integrable structure.

## Contribution

It introduces new compatible Hamiltonian operators for the Krichever-Novikov hierarchy, including compositions of existing operators, enhancing the framework for its integrability analysis.

## Key findings

- H_0, L_4H_0, and L_6H_0 are compatible Hamiltonian operators.
- The hierarchy remains Hamiltonian under these operators.
- The operators extend the known Hamiltonian structure of the equation.

## Abstract

It has been proved by V. Sokolov that the Krichever-Novikov equation's hierarchy is hamiltonian for the non-local Hamiltonian operator H_0=u_x D^{-1} u_x and possesses twi weakly non-local recursion operatos of degree 4 and 6, L_4 and L_6. We show here that H_0, L_4H_0 and L_6H_0 are compatible Hamiltonian operators for which the Krichever-Novikov equation's hierarchy is hamiltonian.