Lenard scheme for two dimensional periodic Volterra chain

TL;DR

This paper establishes a framework for generating infinite hierarchies of symmetries and conservation laws for two-dimensional periodic Volterra chains using Nijenhuis operators, and demonstrates the system's bi-Hamiltonian structure.

Contribution

It constructs a recursion operator from the Lax representation and proves it is a Nijenhuis operator, showing the system's bi-Hamiltonian nature and the relation between Hamiltonian and symplectic operators.

Findings

Constructed a recursion operator from Lax representation.

Proved the recursion operator is Nijenhuis.

Showed the system is bi-Hamiltonian.

Abstract

We prove that for compatible weakly nonlocal Hamiltonian and symplectic operators, hierarchies of infinitely many commuting local symmetries and conservation laws can be generated under some easily verified conditions no matter whether the generating Nijenhuis operators are weakly nonlocal or not. We construct a recursion operator of the two dimensional periodic Volterra chain from its Lax representation and prove that it is a Nijenhuis operator. Furthermore we show this system is a (generalised) bi-Hamiltonian system. Rather surprisingly, the product of its weakly nonlocal Hamiltonian and symplectic operators gives rise to the square of the recursion operator.