Recursion and Hamiltonian operators for integrable nonabelian difference equations
arXiv:1910.06807 · doi:10.1088/1361-6544/aba88c
Abstract
In this paper, we carry out the algebraic study of integrable differential-difference equations whose field variables take values in an associative (but not commutative) algebra. We adapt the Hamiltonian formalism to nonabelian difference Laurent polynomials and describe how to obtain a recursion operator from the Lax representation of an integrable nonabelian differential-difference system. As an application, we propose a novel family of integrable equations: the nonabelian Narita-Itoh-Bogoyavlensky lattice, for which we construct their recursion operators and Hamiltonian operators and prove the locality of infinitely many commuting symmetries generated from their highly nonlocal recursion operators. Finally, we discuss the nonabelian version of several integrable difference systems, including the relativistic Toda chain and Ablowitz-Ladik lattice.
32 pages. Version accepted for publication in Nonlinearity (2020)
References in corpus (23)
- On the local systems Hamiltonian in the weakly nonlocal Poisson brackets
- On Construction of Recursion Operators From Lax Representation
- Darboux transformations and Recursion operators for differential--difference equations
- Reduction Groups and Automorphic Lie Algebras
- Integrable ODEs on Associative Algebras
- Why nonlocal recursion operators produce local symmetries: new results and applications
- On a discrete analog of the Tzitzeica equation
- Lenard scheme for two dimensional periodic Volterra chain
- Double Poisson vertex algebras and non-commutative Hamiltonian equations
- Cosymmetries and Nijenhuis recursion operators for difference equations
- On recursion operators for elliptic models
- Rational recursion operators for integrable differential-difference equations
- Differential-difference equations associated with the fractional Lax operators
- On Construction of Recursion Operator and Algebra of Symmetries for Field and Lattice Systems
- Hamiltonian evolutions of twisted gons in $\RP^n$
- Noncommutative Inverse Scattering Method for the Kontsevich system
- On integrability of the Kontsevich non-abelian ODE system
- On some class of homogeneous polynomials and explicit form of integrable hierarchies of differential-difference equations
- Poisson cohomology of scalar multidimensional Dubrovin-Novikov brackets
- On double Poisson structures on commutative algebras
- On some class of reductions for Itoh-Narita-Bogoyavlenskii lattice
- Local and non-local multiplicative Poisson vertex algebras and differential-difference equations
- A Darboux-Getzler theorem for scalar difference Hamiltonian operators