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SIGMA 2 (2006), 072, 30 pages nlin.SI/0608039
https://doi.org/10.3842/SIGMA.2006.072
Contribution to the Vadim Kuznetsov Memorial Issue
Coupled Modified KP Hierarchy and Its Dispersionless Limit
Takashi Takebe a and Lee-Peng Teo b
a) Department of Mathematics, Ochanomizu University,
Otsuka 2-1-1, Bunkyo-ku, Tokyo, 112-8610, Japan
b) Faculty of Information Technology, Multimedia University,
Jalan Multimedia, Cyberjaya, 63100, Selangor Darul Ehsan, Malaysia
Received August 18, 2006, in final form October 03, 2006; Published online October 31, 2006
Abstract
We define the coupled modified KP hierarchy and its
dispersionless limit. This integrable hierarchy is a generalization of
the ''half'' of the Toda lattice hierarchy as well as an extension of
the mKP hierarchy. The solutions are parametrized by a fibered flag
manifold. The dispersionless counterpart interpolates several versions
of dispersionless mKP hierarchy.
Key words:
cmKP hierarchy; fibered flag manifold; dcmKP hierarchy.
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References
- Chang J.H., Tu M.H., On the Miura map between the
dispersionless
KP and dispersionless modified KP hierarchies, J. Math. Phys., 2000, V.41, 5391-5406,
solv-int/9912016.
- Dickey L.A., Modified KP and discrete KP, Lett. Math. Phys., 1999, V.48, 277-289,
solv-int/9902008.
- Date E., Kashiwara M., Jimbo M., Miwa T., Transformation
groups for soliton equations, in Nonlinear Integrable Systems
- Classical Theory and Quantum Theory, Singapore, World Scientific, 1983, 39-119.
- Duren P.L., Univalent functions, Grundlehren der
Mathematischen Wissenschaften,
Vol. 259, New York, Springer-Verlag, 1983.
- Kashiwara M., Miwa T.,
Transformation groups for soliton equations. I. The t function of
the Kadomtsev-Petviashvili equation, Proc. Japan Acad. Ser. A Math. Sci., 1981, V.57,
342-347.
- Kac V.G., Peterson D.H.,
Lectures on the infinite wedge-representation and the MKP hierarchy,
in Systèmes dynamiques non linéaires: intégrabilité et comportement
qualitatif, Sém. Math. Sup., Vol. 102, Montreal,
Presses Univ. Montréal, 1986, 141-184.
- Kupershmidt B.A.,
KP or mKP. Noncommutative mathematics of Lagrangian,
Hamiltonian, and integrable systems,
Mathematical Surveys and Monographs, Vol. 78,
Providence, RI, American Mathematical Society, 2000.
- Kupershmidt B.A., The quasiclassical limit of the modified
KP
hierarchy, J. Phys. A: Math. Gen., 1990, V.23, 871-886.
- Pommerenke C., Univalent functions, Göttingen, Vandenhoeck & Ruprecht,
1975 (with a chapter on quadratic differentials by Gerd Jensen,
Studia Mathematica/Mathematische Lehrbücher, Band XXV).
- Sato M., Soliton equations as dynamical systems on an infinite
dimensional Grassmann manifold, RIMS Kokyuroku, 1981, V.439,
30-46.
- Sato M., Noumi M., Soliton equations and the universal
Grassmann manifolds, Sophia University Kokyuroku in
Math., Vol. 18, Tokyo, Sophia University, 1984 (in Japanese).
- Sato M., Sato Y., Soliton equations as dynamical systems on
infinite dimensional Grassmann manifold, in Nonlinear Partial
Differential Equations in Applied Science, Proceedings of the
U.S.-Japan Seminar (1982, Tokyo), Lect. Notes in Num. Anal., 1982, V.5, 259-271.
- Takasaki K., Initial value problem for the Toda lattice hierarchy,
Adv. Stud. Pure Math., Vol. 4,
Group Representations and Systems of Differential Equations (1982, Tokyo),
Amsterdam, North-Holland, 1984, 139-163.
- Takebe T., Toda lattice hierarchy and conservation laws,
Comm. Math. Phys., 1990, V.129, 281-318.
- Takebe T., Representation theoretical meaning of the initial value
problem for the Toda lattice hierarchy. I, Lett. Math. Phys., 1991, V.21, 77-84.
- Takebe T., Representation theoretical meaning of the initial value
problem for the Toda lattice hierarchy. II, Publ. RIMS, 1991, V.27, 491-503.
- Takebe T., A note on modified KP hierarchy and its (yet another)
dispersionless limit,
Lett. Math. Phys., 2002, V.59, 157-172,
nlin.SI/0111012.
- Teo L.-P., On dispersionless coupled modified KP hierarchy,
nlin.SI/0304007.
- Teo L.-P., Analytic functions and integrable
hierarchies - characterization of tau functions, Lett. Math. Phys.,
2003, V.64, 75-92,
hep-th/0305005.
- Takasaki K., Takebe T., SDiff(2) KP hierarchy,
in Infinite
Analysis, Part A, B (1991, Kyoto), Adv. Ser. Math. Phys., Vol. 16, River Edge, NJ, World Sci.
Publishing, 1992, 889-922, hep-th/9112046.
- Takasaki K., Takebe T., Integrable hierarchies and
dispersionless
limit, Rev. Math. Phys., 1995, V.7, 743-808, hep-th/9405096.
- Takasaki K., Takebe T., Löwner equations and dispersionless
hierarchies, in the Proceedings of XXIII International
Conference of Differential Geometric Methods in Theoretical Physics
Nankai Institute of Mathematics (August 2005, Tianjin, China), to appear,
nlin.SI/0512008.
- Ueno K., Takasaki K., Toda lattice hierarchy,
Adv. Stud. Pure Math., Vol. 4,
Group Representations and Systems of Differential Equations (1982, Tokyo),
Amsterdam, North-Holland, 1984, 1-95.
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