Lenard scheme for two dimensional periodic Volterra chain

Details Lenard scheme for two dimensional periodic Volterra chain Lenard scheme for two dimensional periodic Volterra chain

2008-09-23

arxiv.org/abs/0809.3899v1

Journal of Mathematical Physics 50(2), 023506, 2009

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Submit to Journal of Mathematical Physics

Scientific paper

10.1063/1.3054921

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.

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