Using the bicomplex approach we discuss an integrable noncommutative system in two-dimensional Euclidean space. It is described by an equation of motion which reduces to the ordinary sine-Gordon equation when the noncommutation parameter is removed, plus a constraint equation which is nontrivial only in the noncommutative case. The implications of this constraint, which is required by integrability but seems to reduce the space of classical solutions, remain to be understood. We show that the system has an infinite number of conserved currents and we give the general recursive relation for constructing them. For the particular cases of lower spin nontrivial currents we work out the explicit expressions and perform a direct check of their conservation. These currents reduce to the usual sine-Gordon currents in the commutative limit. We find classical "localized" solutions to first order in the noncommutativity parameter and describe the Backlund transformations for our system. Finally, we comment on the relation of our noncommutative system to the commutative sine-Gordon system. © 2003 Elsevier Science B.V. All rights reserved.

Grisaru, M., Penati, S. (2003). An Integrable noncommutative version of the sine-Gordon system. NUCLEAR PHYSICS. B, 655(3), 250-276 [10.1016/S0550-3213(03)00064-6].

An Integrable noncommutative version of the sine-Gordon system

PENATI, SILVIA
2003

Abstract

Using the bicomplex approach we discuss an integrable noncommutative system in two-dimensional Euclidean space. It is described by an equation of motion which reduces to the ordinary sine-Gordon equation when the noncommutation parameter is removed, plus a constraint equation which is nontrivial only in the noncommutative case. The implications of this constraint, which is required by integrability but seems to reduce the space of classical solutions, remain to be understood. We show that the system has an infinite number of conserved currents and we give the general recursive relation for constructing them. For the particular cases of lower spin nontrivial currents we work out the explicit expressions and perform a direct check of their conservation. These currents reduce to the usual sine-Gordon currents in the commutative limit. We find classical "localized" solutions to first order in the noncommutativity parameter and describe the Backlund transformations for our system. Finally, we comment on the relation of our noncommutative system to the commutative sine-Gordon system. © 2003 Elsevier Science B.V. All rights reserved.
Articolo in rivista - Articolo scientifico
Integrable systems; Noncommutative geometry; Sine-Gordon;
English
250
276
27
Grisaru, M., Penati, S. (2003). An Integrable noncommutative version of the sine-Gordon system. NUCLEAR PHYSICS. B, 655(3), 250-276 [10.1016/S0550-3213(03)00064-6].
Grisaru, M; Penati, S
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/3268
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