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Dimakis, Aristophanes; Müller‐Hoissen, Folkert

doi:10.1023/a:1021320617277

Cited by 8 publications

(23 citation statements)

References 8 publications

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“…In order to apply this result, we need to know more about WNA algebras. For our purposes, it is sufficient to recall from [1] that any WNA algebra with dim(A/A ′ ) = 1 is isomorphic to one determined by the following data: (1) an associative algebra A (e.g. any matrix algebra) (2) a fixed element g ∈ A (3) linear maps L, R : A → A such that…”

Section: Theoremmentioning

confidence: 99%

From nonassociativity to solutions of the KP hierarchy

Dimakis

Müller‐Hoissen

2006

Czech J Phys

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A recently observed relation between 'weakly nonassociative' algebras A (for which the associator (A, A 2 , A) vanishes) and the KP hierarchy (with dependent variable in the middle nucleus A ′ of A) is recalled. For any such algebra there is a nonassociative hierarchy of ODEs, the solutions of which determine solutions of the KP hierarchy. In a special case, and with A ′ a matrix algebra, this becomes a matrix Riccati hierarchy which is easily solved. The matrix solution then leads to solutions of the scalar KP hierarchy. We discuss some classes of solutions obtained in this way.

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Section: Theoremmentioning

confidence: 99%

From nonassociativity to solutions of the KP hierarchy

Dimakis

Müller‐Hoissen

2006

Czech J Phys

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“…The NC KP hierarchy of the operator form is defined as follows [2,3,17]. We consider an operator valued monic pseudo differential operator (PDO)…”

Section: Noncommutative Kp and Toda Hierarchymentioning

confidence: 99%

“…[2], they derive the differential equations of the deformation parameters θ ij and remark that those equations allow us to construct the solutions of the NC KP hierarchy as formal Taylor series of θ ij from the solutions of the commutative KP hierarchy.…”

Section: A Solution Space Of the Nc Kp Hierarchymentioning

confidence: 99%

“…It seems to imply the significance of the NC deformation of the integrable systems. Some authors have discussed the NC integrable systems on lower dimension [2,3,6,13,17] (See also [5,15,16]), especially the NC KP hierarchy. In this letter, we study the moduli space of the NC KP hierarchy throughout the Birkhoff decomposition of a certain formal group over a NC algebra.…”

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confidence: 99%

“…The deformation parameters θ ij ∈ R are non-zero constants and we assume that a matrix (θ ij ) is invertible. By an orthogonal change of coordinates as [t 2n−1 ,t 2n ] = iθ n (n ≥ 1), the algebra is realized as operators over the Fock spaceThe NC KP hierarchy of the operator form is defined as follows [2,3,17]. We consider an operator valued monic pseudo differential operator (PDO)…”

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confidence: 99%

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Factorization methods for noncommutative KP and Toda hierarchy

Sakakibara¹

2004

J. Phys. A: Math. Gen.

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We consider the solution space of the noncommutative KP hierarchy with the noncommutative extension of the Sato theory. The noncommutative deformation of the Toda hierarchy is introduced. We derive the bilinear identities for the BakerAkhiezer functions and calculate the N -soliton solutions of the noncommutative Toda hierarchy. Noncommutative KP and Toda hierarchyRecently, the classical and quantum field theory over the noncommutative (NC) space times is extensively studied. The NC gauge theory has a great success in the string theory. In particular, the NC deformation of the ADHM construction of the (anti)SDYM equation [9] and the Nahm construction [1] were shown. It seems to imply the significance of the NC deformation of the integrable systems. Some authors have discussed the NC integrable systems on lower dimension [2,3,6,13,17] (See also [5,15,16]), especially the NC KP hierarchy. In this letter, we study the moduli space of the NC KP hierarchy throughout the Birkhoff decomposition of a certain formal group over a NC algebra. We also introduce the NC Toda hierarchy and derive the bilinear identities and the N -soliton solutions.Let (t 1 ,t 2 , · · · ) be coordinates of NC plane R 2∞ which satisfy [t i ,t j ] = iθ ij , and A θ a set of functions on it. The deformation parameters θ ij ∈ R are non-zero constants and we assume that a matrix (θ ij ) is invertible. By an orthogonal change of coordinates as [t 2n−1 ,t 2n ] = iθ n (n ≥ 1), the algebra is realized as operators over the Fock spaceThe NC KP hierarchy of the operator form is defined as follows [2,3,17]. We consider an operator valued monic pseudo differential operator (PDO)

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Generalized Volterra lattices: Binary Darboux transformations and self-consistent sources

Müller‐Hoissen

Chvartatskyi

Toda

2017

Journal of Geometry and Physics

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We study two families of matrix versions of generalized Volterra (or Bogoyavlensky) lattice equations. For each family, the equations arise as reductions of a partial differential-difference equation in one continuous and two discrete variables, which is a realization of a general integrable equation in bidifferential calculus. This allows to derive a binary Darboux transformation and also self-consistent source extensions via general results of bidifferential calculus. Exact solutions are constructed from the simplest seed solutions.

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Abstract: Generalized matrix Lotka-Volterra lattice equations are obtained in a systematic way from a "master equation" possessing a bicomplex formulation.

Cited by 8 publications

References 8 publications

From nonassociativity to solutions of the KP hierarchy

From nonassociativity to solutions of the KP hierarchy

Factorization methods for noncommutative KP and Toda hierarchy

Generalized Volterra lattices: Binary Darboux transformations and self-consistent sources

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