Gauge equivalence theory of the noncompact Ishimori model and the Davey–Stewartson equation

Leo, R. A.; Martina, L.; Soliani, G.

doi:10.1063/1.529676

Cited by 18 publications

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“…This formalism arises as a natural generalization of the gauge equivalence approach developed for the 1+1 dimensional Heisenberg model, which in this way is mapped into the Nonlinear Schrödinger Equation (NLSE) [9]. A similar correspondence [10] can be established between the Ishimori model and the DaveyStewartson equation, which are the integrable extensions of the previous models in (2+1)-dimensions. These mappings are essentially given by the chiral current on a Lie algebra, determined by the original system, and such relations can be established independently from the Lax representation and the integrability properties.…”

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confidence: 99%

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Chern-Simons gauge field theory of two-dimensional ferromagnets

Martina

Pashaev

Soliani³

1993

Phys. Rev. B

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A Chern-Simons gauged Nonlinear Schrödinger Equation is derived from the continuous Heisenberg model in 2+1 dimensions. The corresponding planar magnets can be analyzed whithin the anyon theory. Thus, we show that static magnetic vortices correspond to the self-dual Chern -Simons solitons and are described by the Liouville equation. The related magnetic topological charge is associated with the electric charge of anyons. Furthermore, vortex -antivortex configurations are described by the sinh-Gordon equation and its conformally invariant extension. Physical consequences of these results are discussed.Lecce, June 1993 1

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confidence: 99%

“…Equation (12) allows us to eliminate the function q 0 from the zero curvature conditions (9)(10), obtaining in such a way the gauge invariant evolution equations…”

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Chern-Simons gauge field theory of two-dimensional ferromagnets

Martina

Pashaev

Soliani³

1993

Phys. Rev. B

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“…The linear problem of IM can be gauge related with the one of the DS equation [15,16]. And the last linear problem can be reduced from the self-dual Yang-Mills system with an infinite dimensional gauge group [17].…”

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confidence: 99%

Integrable Chern-Simons Gauge Field Theory in 2+1 Dimensions

Pashaev

1996

Mod. Phys. Lett. A

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The classical spin model in planar condensed media is represented as the U (1) Chern-Simons gauge field theory. When the vorticity of the continuous flow of the media coincides with the statistical magnetic field, which is necessary for the model's integrability, the theory admits zero curvature connection. This allows me to formulate the model in terms of gauge -invariant fields whose evolution is described by In the present paper I reduce the self-dual CS model from an integrable model in 2+1 dimensions. This model was first derived by Ishimori and has integrable dynamics of the magnetic vortices [7]. Its formulation in terms of the vorticity for continuous flow tends to clarify the physical meaning of the model and allows an extension for higher dimensions, admitting the bilinear Hirota representation [8,10]. The physical applications of the model are connected with the resolution of the anomalous behavior of the linear momentum in ferromagnets [9]. As it was shown in the latter paper by using the de-localized electron model of ferromagnets, the canonical momentum becomes well defined due to a fermionic background.The model considered in the present paper is a modification of the classical Heisenberg model for condensed media, having hydrodynamical flow with its vorticity related to a topological charge density. In this case a hydrodynamical vortex in the magnetic (anyon) fluid is also inducing a magnetic vorticity.The CS gauge field theory is then obtained by projecting spin variables on the tangent plane to the sphere (spin phase space). This idea has been recently applied to the classical Heisenberg model in [12]. Furthermore, the stationary magnetic vortices of the model have been related to the Chern-Simons solitons, while the topological charge of the former has been related to the electric charge of the latter. 2In the case of equality between hydrodynamical vorticity and the "statistical" magnetic field, the theory can be formulated for a zero curvature effective gauge field. This allows me to reformulate the model in terms of gauge invariant matter fields. As shown, their evolution is described by the DS equations and at the same time the fields satisfy the CS system. For the self-dual case, we reproduce CS solitons with an integrable evolution replacing the statistical gauge field with a velocity field.As a by-product we obtain: a) new reduction conditions for the DS-I model related to the Ishimori-I (IM-I) model, b) the 2+1 dimensional zero curvature formulation for the DS equation. Zero Curvature ConditionThe crucial moment of our construction is a zero curvature condition in 2+1 dimensional spacefor u(2)(u(1, 1)) Lie algebra valued connection J µ , (µ = 0, 1, 2). Decomposing the algebra for the diagonal and off-diagonal partsand parametrizingin terms of two U (1) gauge potentials W µ and V µ , and the q µ matter fields, we have the set of U (1) × U (1) gauge invariant equationsHere the covariant derivative is defined asFor the spatial part of the matter fields we introduceand denote D ± =...

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“…Ishimori provided a Lax pair and derived multivortex solutions using the Hirota method, thus showing that the vortex dynamics is integrable. A noncompact version of the Ishimori spin model and the Davey-Stewartson equation were shown to be gauge equivalent in [13].…”

Section: Bäcklund Algebras For Ishimori Modelsmentioning

confidence: 98%