It is well-known that the general Manakov system is a 2-components nonlinear Schrodinger equation with 4 nonzero real parameters.The analytic property of the general Manakov system has been well-understood though it l...It is well-known that the general Manakov system is a 2-components nonlinear Schrodinger equation with 4 nonzero real parameters.The analytic property of the general Manakov system has been well-understood though it looks complicated.This paper devotes to exploring geometric properties of this system via the prescribed curvature representation in the category of Yang-Mills’theory.Three models of moving curves evolving in the symmetric Lie algebras u(2,1)=k_(α)⊕m_(α)(α=1,2)and u(3)=k_(3)⊕m_(3)are shown to be simultaneously the geometric realization of the general Manakov system.This reflects a new phenomenon in geometric realization of a partial differential equation/system.展开更多
By using holomorphic Riemannian geometry in C^3, the coupled Landau-Lifshitz(CLL) equation is proved to be exactly the equation of Schr¨odinger flows from R^1 to the complex 2-sphere CS^2(1) → C^3.Furthermore, r...By using holomorphic Riemannian geometry in C^3, the coupled Landau-Lifshitz(CLL) equation is proved to be exactly the equation of Schr¨odinger flows from R^1 to the complex 2-sphere CS^2(1) → C^3.Furthermore, regarded as a model of moving complex curves in C^3, the CLL equation is shown to preserve the PT symmetry if the initial data is of the P symmetry. As a consequence, the nonlocal nonlinear Schrodinger(NNLS)equation proposed recently by Ablowitz and Musslimani is proved to be gauge equivalent to the CLL equation with initial data being restricted by the P symmetry. This gives an accurate characterization of the gaugeequivalent magnetic structure of the NNLS equation described roughly by Gadzhimuradov and Agalarov(2016).展开更多
In this article,we devote to a mathematical survey on the theory of the vortex filament in 3-dimensional spaces and its generalizations.We shall present some effective geometric tools applied in the study,such as the ...In this article,we devote to a mathematical survey on the theory of the vortex filament in 3-dimensional spaces and its generalizations.We shall present some effective geometric tools applied in the study,such as the Schrodinger flow,the geometric Korteweg-de Vries(KdV)flow and the generalized bi-Schrodinger flow,as well as the complex and para-complex structures.It should be mentioned that the investigation in the imaginary part of the octonions looks very fascinating,since it relates to almost complex structures and the G2 structure on S^(6).As a new result in this survey,we describe the equation of generalized bi-Schr?dinger flows from R1 into a Riemannian surface.展开更多
基金supported by the National Natural Science Foundation of China(Nos.12071080,12141104)the Science Technology Project of Jiangxi Educational Committee(No.GJJ2201202)the Natural Science Foundation of Jiangxi Province(Nos.20212BAB211005,20232BAB201006)。
文摘It is well-known that the general Manakov system is a 2-components nonlinear Schrodinger equation with 4 nonzero real parameters.The analytic property of the general Manakov system has been well-understood though it looks complicated.This paper devotes to exploring geometric properties of this system via the prescribed curvature representation in the category of Yang-Mills’theory.Three models of moving curves evolving in the symmetric Lie algebras u(2,1)=k_(α)⊕m_(α)(α=1,2)and u(3)=k_(3)⊕m_(3)are shown to be simultaneously the geometric realization of the general Manakov system.This reflects a new phenomenon in geometric realization of a partial differential equation/system.
基金supported by National Natural Science Foundation of China(Grant No.11271073)。
文摘By using holomorphic Riemannian geometry in C^3, the coupled Landau-Lifshitz(CLL) equation is proved to be exactly the equation of Schr¨odinger flows from R^1 to the complex 2-sphere CS^2(1) → C^3.Furthermore, regarded as a model of moving complex curves in C^3, the CLL equation is shown to preserve the PT symmetry if the initial data is of the P symmetry. As a consequence, the nonlocal nonlinear Schrodinger(NNLS)equation proposed recently by Ablowitz and Musslimani is proved to be gauge equivalent to the CLL equation with initial data being restricted by the P symmetry. This gives an accurate characterization of the gaugeequivalent magnetic structure of the NNLS equation described roughly by Gadzhimuradov and Agalarov(2016).
基金The first author was supported by National Natural Science Foundation of China(Grant Nos.11531012,11926307 and 12071080).
文摘In this article,we devote to a mathematical survey on the theory of the vortex filament in 3-dimensional spaces and its generalizations.We shall present some effective geometric tools applied in the study,such as the Schrodinger flow,the geometric Korteweg-de Vries(KdV)flow and the generalized bi-Schrodinger flow,as well as the complex and para-complex structures.It should be mentioned that the investigation in the imaginary part of the octonions looks very fascinating,since it relates to almost complex structures and the G2 structure on S^(6).As a new result in this survey,we describe the equation of generalized bi-Schr?dinger flows from R1 into a Riemannian surface.
