Integrable Systems in Symplectic Geometry

Asadi, Esmaeel; Sanders, Jan A.

doi:10.1142/9789812776174_0003

Cited by 3 publications

(4 citation statements)

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“…Such a zero-curvature approach has been used in recent work [32] to derive a quaternionic mKdV equation for the variable u = ω x = ω⌋γ x , based on the choice of a quaternionic connection matrix given by This differs compared to our choice given by a parallel framing (3.12) and leads to a more complicated bi-Hamiltonian structure [32] …”

Section: Discussionmentioning

confidence: 99%

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Quaternionic soliton equations from Hamiltonian curve flows in {\bb H}{\bb P}^n

Anco¹,

Asadi²

2009

J. Phys. A: Math. Theor.

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Abstract. A bi-Hamiltonian hierarchy of quaternion soliton equations is derived from geometric non-stretching flows of curves in the quaternionic projective space HP n . The derivation adapts the method and results in recent work by one of us on the Hamiltonian structure of non-stretching curve flows in Riemannian symmetric spacesas a symmetric space in terms of compact real symplectic groups and quaternion unitary groups. As main results, scalar-vector (multi-component) versions of the sine-Gordon (SG) equation and the modified Korteweg-de Vries (mKdV) equation are obtained along with their bi-Hamiltonian integrability structure consisting of a shared hierarchy of quaternionic symmetries and conservation laws generated by a hereditary recursion operator. The corresponding geometric curve flows in HP n are shown to be described by a non-stretching wave map and a mKdV analog of a non-stretching Schrödinger map.

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Section: Discussionmentioning

confidence: 99%

“…However, it was not shown in [32] whether the above choice for the connection matrix can be achieved by a gauge transformation starting from an arbitrary form of u = ω x ∈ h.…”

Section: Discussionmentioning

confidence: 99%

Quaternionic soliton equations from Hamiltonian curve flows in {\bb H}{\bb P}^n

Anco¹,

Asadi²

2009

J. Phys. A: Math. Theor.

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“…The research was done by the first author, supported by N WO, under supervision of the second and published in [3].…”

Section: Introductionmentioning

confidence: 99%

“…We will give details of the proof of some of the assertions. The reader can find the missing details in [3]. a general manifold a choice of a connection prescribes a way of translating tangent vectors 'parallel to themselves' and to intrinsically define a directional derivative.…”

Section: Introductionmentioning

confidence: 99%

Integrable Systems in Symplectic Geometry

Asadi

Sanders

2009

Glasgow Math. J.

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Abstract. Quaternionic vector mKDV equations are derived from the Cartan structure equation in the symmetric space ‫ސވ‬ n = Sp(n + 1)/Sp(1) × Sp(n). The derivation of the soliton hierarchy utilizes a moving parallel frame and a Cartan connection 1-form ω related to the Cartan geometry on ‫ސވ‬ n modelled on (sp n+1 , sp 1 × sp n ). The integrability structure is shown to be geometrically encoded by a PoissonNijenhuis structure and a symplectic operator.

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Symplectically invariant soliton equations from non-stretching geometric curve flows

Anco¹,

Asadi²

2012

J. Phys. A: Math. Theor.

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department of mathematics, brock university, st. catharines, on canada department of mathematics, institute for advanced studies in basic sciences, gava zang, zanjan, iranAbstract. Bi-Hamiltonian hierarchies of symplectically-invariant soliton equations are derived from geometric non-stretching flows of curves in the Riemannian symmetric spaces Sp(n+1)/Sp(1)×Sp(n) and SU (2n)/Sp(n). The derivation uses Hasimoto variables defined by a moving parallel frame along the curves. As main results, two new multi-component versions of the sine-Gordon (SG) equation and the modified Korteweg-de Vries (mKdV) equation exhibiting Sp(1) × Sp(n − 1) invariance are obtained along with their bi-Hamiltonian integrability structure consisting of a hierarchy of symmetries and conservation laws generated by a hereditary recursion operator. The corresponding geometric curve flows in both Sp(n + 1)/Sp(1) × Sp(n) and SU (2n)/Sp(n) are shown to be described by a non-stretching wave map and a mKdV analog of a non-stretching Schrödinger map.8

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Integrable Systems in Symplectic Geometry

Cited by 3 publications

References 10 publications

Quaternionic soliton equations from Hamiltonian curve flows in {\bb H}{\bb P}^n

Quaternionic soliton equations from Hamiltonian curve flows in {\bb H}{\bb P}^n

Integrable Systems in Symplectic Geometry

Symplectically invariant soliton equations from non-stretching geometric curve flows

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