Integrable generalizations of Schrödinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces

Anco, Stephen C.; Myrzakulov, Ratbay

doi:10.1016/j.geomphys.2010.05.013

Cited by 52 publications

(28 citation statements)

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“…which is shown to be skew-symmetric by using (10). Furthermore, in a similar way to the proof of Theorem 2, we see that ω m+1 is a presymplectic form and X n is a Hamiltonian vector field for the Hamiltonian function H n+m+1 with respect to ω m+1 ; indeed, for functions F , G given by (7) and for an integer k, putting…”

Section: Multi-hamiltonian Structures On the Level Sets Of Hamiltonianssupporting

confidence: 62%

“…On the other hand, it is known that a lot of completely integrable systems are described as bi-Hamiltonian systems, from which the existence of many first integrals can be deduced (Magri's theorem [22,27]). In this context, many of motions of curves as above have been studied from the viewpoint of bi-Hamiltonian systems recently [1,2,3,4,5,6,7,8,21,23,24,31]. The purpose of this paper is to construct a multi-Hamiltonian structure associated to the higher KdV flows on each level set of Hamiltonian functions in a geometric way (Theorem 7).…”

Section: Introductionmentioning

confidence: 99%

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Multi-Hamiltonian Structures on Spaces of Closed Equicentroaffine Plane Curves Associated to Higher KdV Flows

Fujioka

Kurose

2014

SIGMA

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Section: Multi-hamiltonian Structures On the Level Sets Of Hamiltonianssupporting

confidence: 62%

Section: Introductionmentioning

confidence: 99%

Multi-Hamiltonian Structures on Spaces of Closed Equicentroaffine Plane Curves Associated to Higher KdV Flows

Fujioka

Kurose

2014

SIGMA

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“…The equivalence between the scalar NLS equation u t = −i(u xx + 2|u| 2 u), the Schrödinger map equation into S 2 , and the Heisenberg spin model in su(2) ≃ R 3 , as well as their integrability structures, is discussed in Ref. [12].…”

Section: Integrable Systems Arising From a Complex Vector-valued Hasimentioning

confidence: 99%

Hasimoto variables, generalized vortex filament equations, Heisenberg models and Schrödinger maps arising from group-invariant NLS systems

Anco

Asadi

2019

Journal of Geometry and Physics

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1 department of mathematics and statistics brock university st. catharines, on l2s3a1, canada 2 department of mathematics institute for advance studies in basic science (iasbs) 45137-66731, zanjan, iran Abstract. The deep geometrical relationships holding among the NLS equation, the vortex filament equation, the Heisenberg spin model, and the Schrödinger map equation are extended to the general setting of Hermitian symmetric spaces. New results are obtained by utilizing a generalized Hasimoto variable which arises from applying the general theory of parallel moving frames. The example of complex projective space CP N = SU (N + 1)/U (N ) is used to illustrate the method and results.of the curve, while u = κe i τ dx has the geometrical meaning of a U(1)-covariant of the curve, with (Re u, Im u) being the components of the Cartan matrix of a parallel frame [18] given by the tangent vector T and the pair of normal vectors Re(e i τ dx (N + iB)), Im(e i τ dx (N + iB)). The phase-rotation symmetry on u corresponds to a U(1) ≃ SO(2) gauge group of transformations in SO(3) preserving the structure of a parallel frame by rigid rotations in the normal plane along the curve. As shown by Hasimoto [22], the bi-normal equation also describes the physical motion of a vortex filament in fluid mechanics. Furthermore, the bi-normal equation can be interpreted as a Heisenberg spin model in R 3 , or equivalently as a Schrödinger map equation on the sphere S 2 . The geometrical interrelationships among these physical and mathematical formulations of the NLS equation have been explored in Ref. [12]. Abstract formulations involving differential invariants, Hamiltonian structures, and loop groups can be found in Refs. [33,15,16].A generalization of linear isospectral flows yielding group invariant multi-component integrable NLS systems is known [19] for all Lie algebras associated with Hermitian symmetric spaces (namely, Riemannian symmetric spaces with a compatible complex structure). These isospectral flows, called Fordy-Kulish systems, each have an associated Sym-Pohlmeyer curve flow which can be viewed as a Lie-algebra version of a bi-normal equation [25].A generalization of parallel frames and Hasimoto variables yielding group-invariant integrable systems is known [4] for all Riemannian symmetric spaces. This includes [3] semisimple Lie groups, viewed in a natural way as diagonal Riemannian symmetric spaces. In particular, the results in Ref. [3] show that the NLS equation arises directly from a parallel frame formulation of a non-stretching geometric curve flow given by a chiral Schrödinger map equation in the Lie group SU(2), which is a variant of the Sym-Pohlmeyer curve flow in the Lie algebra su(2). This type of geometrical derivation of group-invariant integrable systems and corresponding non-stretching map equations has been pursued [30,2,5,6,7,8] for all of the basic classical (non-exceptional) Riemannian symmetric spaces, yielding many types of multi-component mKdV and sine-Gordon systems, as well as multi-component NLS syst...

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“…Here T = R/2πZ is the one-dimensional flat torus, u = u(t, x) : R × T → N is the unknown map describing the deformation of closed curves lying on N parameterized by t, u 0 = u 0 (x) : 1 T → N is the given initial map, u t = du( ∂ ∂t ), u x = du( ∂ ∂x ), du is the differential of the map u, ∇ x is the covariant derivative along u in x, J u : T u N → T u N is the complex structure at u ∈ N, a, b, c, and λ are real constants. If a, b, c = 0 and λ = 1, (1.1) is reduced to the second-order dispersive equation of the form…”

Section: )mentioning

confidence: 99%

A Fourth-Order Dispersive Flow Equation for Closed Curves on Compact Riemann Surfaces

Onodera

2017

J Geom Anal

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A fourth-order dispersive flow equation for closed curves on the canonical twodimensional unit sphere arises in some contexts in physics and fluid mechanics. In this paper, a geometric generalization of the sphere-valued model is considered, where the solutions are supposed to take values in compact Riemann surfaces. As a main results, time-local existence and the uniqueness of a solution to the initial value problem is established under the assumption that the sectional curvature of the Riemann surface is constant. The analytic difficulty comes from the so-called loss of derivatives and the absence of the local smoothing effect. The proof is based on the geometric energy method combined with a kind of gauge transformation to eliminate the loss of derivatives. Specifically, to show the uniqueness of the solution, the detailed geometric analysis of the solvable structure for the equation is presented.2000 Mathematics Subject Classification. Primary 53C44; Secondary 35Q35, 35Q55, 35G61. Key words and phrases. dispersive flow, geometric analysis, local existence and uniqueness, loss of derivatives, energy method, gauge transformation, constant sectional curvature. some conservation laws of (1.4) to show the local existence of a weak solution to the initial value problem, though the uniqueness was unsolved. Chihara in [4] investigated fourth-order dispersive systems for C 2 -valued functions including a system which is reduced from (1.1) by the generalized Hasimoto transformation, and pointed out that the assumption that the sectional curvature of N is constant provides the solvable structure of the initial value problem. To the present author's knowledge, though the insights seems to grasp the solvable structure of (1.1)-(1.2) essentially, it is nontrivial whether we can recover the solution to (1.1)-(1.2) from the solution to the reduced dispersive system.Motivated by them, the present author tried to solve directly (1.1)-(1.2) imposing that the sectional curvature on N is constant, without using the generalized Hasimoto transformation. Recently, he in [18] succeeded to show the local existence of a unique solution to the initial value problem for the S 2 -valued model (1.4) without any assumption on a, b, c, λ (except for a = 0), where u 0 is taken so that u 0x ∈ H k (T; R 3 ) with k 6. This is proved by the energy method based on the standard Sobolev norm for R 3 -valued functions, combined with a kind of gauge transformation.The purpose of the present paper is to extend the results obtained in [18] for S 2 -valued model (1.4), that is, to establish the time-local existence and uniqueness theorem for (1.1)-(1.2) under the assumption that k 6 and the sectional curvature on (N, g) is constant. More precisely, our main results is stated as follows:Theorem 1.

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Integrable generalizations of Schrödinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces

Cited by 52 publications

References 27 publications

Multi-Hamiltonian Structures on Spaces of Closed Equicentroaffine Plane Curves Associated to Higher KdV Flows

Multi-Hamiltonian Structures on Spaces of Closed Equicentroaffine Plane Curves Associated to Higher KdV Flows

Hasimoto variables, generalized vortex filament equations, Heisenberg models and Schrödinger maps arising from group-invariant NLS systems

A Fourth-Order Dispersive Flow Equation for Closed Curves on Compact Riemann Surfaces

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