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Hamiltonian Flows of Curves in G/SO(N) and Vector Soliton Equations of mKdV and Sine-Gordon Type
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2006, том 2
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Hamiltonian Flows of Curves in G/SO(N) and Vector Soliton Equations of mKdV and Sine-Gordon Type
Інший ID:
2000 Mathematics Subject Classification: 37K05; 37K10; 37K25; 35Q53; 53C35
Посилання:
Hamiltonian Flows of Curves in G/SO(N) and Vector Soliton Equations of mKdV and Sine-Gordon Type / S.C. Anco // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 30 назв. — англ.
Підтримка:
I am grateful to Thomas Wolf and Jing Ping Wang for stimulating discussions in motivating this research. I also thank the referees for many valuable comments. Tom Farrar is thanked for assistance with typesetting this paper. The author acknowledges support by an N.S.E.R.C. grant.
Дата:
2006
Переглядів:
822
Завантажень:
274
Анотація:
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.
