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SIGMA, 2006, том 2, 044, 18 страниц (Mi sigma72)  

Эта публикация цитируется в 30 научных статьях (всего в 30 статьях)

Hamiltonian Flows of Curves in $G/SO(N)$ and Vector Soliton Equations of mKdV and Sine-Gordon Type

Stephen C. Anco

Department of Mathematics, Brock University, Canada

Аннотация: The bi-Hamiltonian structure of the two known vector generalizations of the mKdV hierarchy of soliton equations is derived in a geometrical fashion from flows of non-stretching curves in Riemannian symmetric spaces $G/SO(N)$. These spaces are exhausted by the Lie groups $G=SO(N+1),SU(N)$. The derivation of the bi-Hamiltonian structure uses a parallel frame and connection along the curve, tied to a zero curvature Maurer–Cartan form on $G$, and this yields the mKdV recursion operators in a geometric vectorial form. The kernel of these recursion operators is shown to yield two hyperbolic vector generalizations of the sine-Gordon equation. The corresponding geometric curve flows in the hierarchies are described in an explicit form, given by wave map equations and mKdV analogs of Schrödinger map equations.

Ключевые слова: bi-Hamiltonian; soliton equation; recursion operator; symmetric space; curve flow; wave map; Schrödinger map; mKdV map


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Реферативные базы данных:

ArXiv: nlin.SI/0512046
Тип публикации: Статья
MSC: 37K05; 37K10; 37K25; 35Q53; 53C35
Поступила: 12 декабря 2005 г.; в окончательном варианте 12 апреля 2006 г.; опубликована 19 апреля 2006 г.
Язык публикации: английский

Образец цитирования: Stephen C. Anco, “Hamiltonian Flows of Curves in $G/SO(N)$ and Vector Soliton Equations of mKdV and Sine-Gordon Type”, SIGMA, 2 (2006), 044, 18 pp.

Цитирование в формате AMSBIB
\by Stephen C.~Anco
\paper Hamiltonian Flows of Curves in $G/SO(N)$ and Vector Soliton Equations of mKdV and Sine-Gordon Type
\jour SIGMA
\yr 2006
\vol 2
\papernumber 044
\totalpages 18

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    Эта публикация цитируется в следующих статьяx:
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    2. Changzheng Qu, Yanyan Li, “Deformation of Surfaces Induced by Motions of Curves in Higherdimensional Similarity Geometries”, Methods and Applications of Analysis, 14:3 (2007), 273–286  crossref  mathscinet
    3. Gloria Marí Beffa, “Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems”, SIGMA, 4 (2008), 034, 23 pp.  mathnet  crossref  mathscinet  zmath
    4. Qu, CZ, “Higher-Dimensional Integrable Systems Arising from Motions of Curves on S-2(R) and S-3(R)”, Communications in Theoretical Physics, 50:4 (2008), 841  crossref  mathscinet  adsnasa  isi  scopus
    5. Li, YY, “Higher-dimensional integrable systems induced by motions of curves in affine geometries”, Chinese Physics Letters, 25:6 (2008), 1931  crossref  adsnasa  isi  elib  scopus
    6. Anco, SC, “Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces”, Journal of Geometry and Physics, 58:1 (2008), 1  crossref  mathscinet  zmath  adsnasa  isi  scopus
    7. Anco S.C., “Hamiltonian Curve Flows in Lie Groups G Subset of U(N) and Vector NLS, Mkdv, sine-Gordon Soliton Equations”, Symmetries and Overdetermined Systems of Partial Differential Equations, IMA Volumes in Mathematics and its Applications, 144, eds. Eastwood M., Miller W., Springer, 2008, 223–250  crossref  mathscinet  zmath  isi
    8. Beffa G.M., “Projective-Type Differential Invariants for Curves and their Associated PDEs of KdV Type”, Symmetries and Overdetermined Systems of Partial Differential Equations, IMA Volumes in Mathematics and its Applications, 144, eds. Eastwood M., Miller W., Springer, 2008, 265–275  crossref  mathscinet  zmath  isi
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