An integrable noncommutative version of the sine-Gordon system

Abstract

Using the bicomplex approach we discuss a 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. 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.

Integrable noncommutative sine-Gordon model

“…For the case at hand, for example, different inequivalent ansätze for the U(2) matrices entering the bicomplex construction [13] are possible as long as they all reproduce the ordinary sine-Gordon equation in the commutative limit. It is therefore conceivable that among these possibilities there exists an ansatz (different from the one in [22,17]) which guarantees the classical integrability of the corresponding noncommutative model. What is already certain is the necessity to introduce two real scalar fields instead of one, since in the noncommutative realm the U(1) subgroup of U(2) fails to decouple.…”

Interaction of noncommutative solitons with defects

“…Integrability problem can be solved by using the bicomplex implemented method in noncommutative geometry for the NCSG model, which has been fully demonstrated in [6]. The results are unexpected and differ from the 'natural' generalized NCSG model.…”

“…φ n θ n (6) Note that the dependency to the parameter θ arises from the expansion (6) and also from the definition of *-product in the equations.…”

“…For this situation equations (4) reduce to [6] ∂∂φ 0 = λsinφ 0 ∂∂φ 1 = λφ 1 cosφ 0 (7) and the constraint equation…”

Noncommutative vortices and instantons from generalized Bose operators

“…Detailed investigations of noncommutative gauge theories have led to the discovery of localized static classical solutions in noncommutative spaces [14][15][16][17][18][19][20][21][22]. Among the models of gauge theories in noncommutative spaces, one of the simplest is the Abelian-Higgs model possessing vortex-like solutions [23][24][25][26].…”