Bi-Hamiltonian structure of the extended noncommutative Toda hierarchy

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 study a 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.

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“…The second Hamiltonian structure H 2 , obtained by H 2 = RH 1 , is given by a matrix with the following entries [30]:…”

Section: Nonabelian Todamentioning

confidence: 99%

Recursion and Hamiltonian operators for integrable nonabelian difference equations

Casati

Wang

2020

Nonlinearity

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“…The second Hamiltonian structure H 2 , obtained by H 2 = ℜH 1 , is given by a matrix with the following entries [26]:…”

Section: Nonabelian Volterra the Nonabelian Volterra Chainmentioning

confidence: 99%

Recursion and Hamiltonian operators for integrable nonabelian difference equations

Casati,

Wang

2019

Preprint

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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.

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“…The noncommutative field theory is a fruitful subject in both mathematics and physics particularly in noncommutative integrable systems [18,27]. The noncommutative theory gives rise to various new physical objects in quantum mechanics such as the canonical commutation relation [q, p] = ih.…”

Section: Introductionmentioning

confidence: 99%

Additional symmetries and string equations of the noncommutative B and C type KP hierarchies

Liu¹,

Li²

2021

JNMP

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In this paper, we construct the noncommutative B and C type KP hierarchies using pseudo-differential operators and reducing conditions. Further a series of additional flows of the noncommutative B and C type KP hierarchies will be defined and the additional symmetries constitute the B and C type infinite dimensional Lie algebra W 1+∞ . In addition, the generating function of the additional symmetries can also be proved to have a nice form in terms of wave functions. Further, the string equations of the noncommutative B and C type KP hierarchies are derived.

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Bi-Hamiltonian structure of the extended noncommutative Toda hierarchy

Cited by 6 publications

References 26 publications

Recursion and Hamiltonian operators for integrable nonabelian difference equations

Recursion and Hamiltonian operators for integrable nonabelian difference equations

Recursion and Hamiltonian operators for integrable nonabelian difference equations

Additional symmetries and string equations of the noncommutative B and C type KP hierarchies

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