|
SIGMA 2 (2006), 044, 18 pages nlin.SI/0512046
https://doi.org/10.3842/SIGMA.2006.044
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
Received December 12, 2005, in final form April 12, 2006; Published online April 19, 2006;
Replaced by the revised version September 29, 2006
Abstract
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.
Key words:
bi-Hamiltonian; soliton equation; recursion operator; symmetric space; curve flow; wave map; Schrödinger map; mKdV map.
pdf (294 kb)
ps (202 kb)
tex (21 kb)
References
- Anco S.C.,
Conservation laws of scaling-invariant field equations,
J. Phys. A: Math. Gen., 2003, V.36, 8623-8638, math-ph/0303066.
- Anco S.C.,
Bi-Hamiltonian operators, integrable flows of curves using moving frames,
and geometric map equations,
J. Phys. A: Math. Gen., 2006, V.39, 2043-2072, nlin.SI/0512051.
- Anco S.C., in preparation.
- Anco S.C., Wang J.-P., in preparation.
- Anco S.C., Wolf T.,
Some symmetry classifications of hyperbolic vector evolution equations,
J. Nonlinear Math. Phys., 2005, V.12, suppl. 1, 13-31,
Erratum, J. Nonlinear Math. Phys., 2005, V.12, 607-608, nlin.SI/0412015.
- Athorne C., Fordy A.,
Generalised KdV and mKdV equations associated with symmetric spaces,
J. Phys. A: Math. Gen., 1987, V.20, 1377-1386.
- Athorne C.,
Local Hamiltonian structures of multicomponent KdV equations,
J. Phys. A: Math. Gen., 1988, V.21, 4549-4556.
- Bakas I., Park Q.-H., Shin H.-J.,
Lagrangian formulation of symmetric space sine-Gordon models,
Phys. Lett. B, 1996, V.372, 45-52, hep-th/9512030.
- Bishop R.,
There is more than one way to frame a curve,
Amer. Math. Monthly, 1975, V.82, 246-251.
- Chou K.-S., Qu C.,
Integrable equations arising from motions of plane curves,
Phys. D, 2002, V.162, 9-33.
- Chou K.-S., Qu C.,
Integrable motion of space curves in affine geometry,
Chaos Solitons Fractals, 2002, V.14, 29-44.
- Chou K.-S., Qu C.,
Integrable equations arising from motions of plane curves. II,
J. Nonlinear Sci., 2003, V.13, 487-517.
- Chou K.-S., Qu C.,
Motion of curves in similarity geometries and Burgers-mKdV hierarchies,
Chaos Solitons Fractals, 2004, V.19, 47-53.
- Dorfman I.,
Dirac structures and integrability of nonlinear evolution equations,
Wiley, 1993.
- Foursov M.V.,
Classification of certain integrable coupled potential KdV
and modified KdV-type equations,
J. Math. Phys., 2000, V.41, 6173-6185.
- Helgason S.,
Differential geometry, Lie groups, and symmetric spaces,
Providence, Amer. Math. Soc., 2001.
- Kobayashi S., Nomizu K.,
Foundations of differential geometry, Vols. I and II,
Wiley, 1969.
- Olver P.J.,
Applications of Lie groups to differential equations,
New York, Springer, 1986.
- Pohlmeyer K., Rehren K.-H.,
Reduction of the two-dimensional O(n) nonlinear s-model,
J. Math. Phys., 1979, V.20, 2628-2632.
- Mari Beffa G., Sanders J., Wang J.-P.,
Integrable systems in three-dimensional Riemannian geometry,
J. Nonlinear Sci., 2002, V.12, 143-167.
- Sanders J., Wang J.-P.,
Integrable systems in n dimensional Riemannian geometry,
Mosc. Math. J., 2003, V.3, 1369-1393.
- Sanders J., Wang J.-P., J. Difference Equ. Appl., 2006, to appear.
- Sergyeyev A.,
The structure of cosymmetries and a simple proof of locality for
hierarchies of symmetries of odd order evolution equations,
in Proceedings of Fifth International Conference "Symmetry in
Nonlinear Mathematical Physics" (June 23-29, 2003, Kyiv),
Editors A.G. Nikitin, V.M. Boyko, R.O. Popovych and I.A. Yehorchenko, Proceedings of Institute
of Mathematics, Kyiv,
2004, V.50, Part 1, 238-245.
- Sergyeyev A.,
Why nonlocal recursion operators produce local symmetries:
new results and applications,
J. Phys. A: Math. Gen., 2005, V.38, 3397-3407, nlin.SI/0410049.
- Sergyeyev A., Demskoi D.,
The Sasa-Satsuma (complex mKdV II) and the complex sine-Gordon II equation
revisited: recursion operators, nonlocal symmetries and more,
nlin.SI/0512042.
- Sharpe R.W.,
Differential geometry, New York,
Springer-Verlag, 1997.
- Sokolov V.V., Wolf T.,
Classification of integrable vector polynomial evolution equations,
J. Phys. A: Math. Gen., 2001, V.34, 11139-11148.
- Tsuchida T., Wolf T.,
Classification of polynomial integrable systems of mixed scalar and vector
evolution equations. I,
J. Phys. A: Math. Gen., 2005, V.38, 7691-7733, nlin.SI/0412003.
- Wang J.-P.,
Symmetries and conservation laws of evolution equations,
PhD Thesis, Vrije Universiteit, Amsterdam, 1998.
- Wang J.-P., Generalized Hasimoto transformation and vector sine-Gordon equation,
in SPT 2002: Symmetry and Perturbation Theory (Cala Gonone),
Editors S. Abenda, G. Gaeta and S. Walcher, River Edge, NJ,
World Scientific, 2002, 276-283.
|
|