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From the KP hierarchy to thFrom the KP hierarchy to the Painlevé equationse Painlevé equations
Saburo KAKEI (Rikkyo University)
Joint work with Tetsuya KIKUCHI (University of Tokyo)
Painlevé Equations and Monodromy Problems: Recent Developments
22 September 2006
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Known Facts
Fact 1Painlevé equations can be obtained as similarity reduction of soliton equations.
Fact 2Many (pahaps all) soliton equations can be obtained as reduced cases of Sato’s KP hierarchy.
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Similarity
Similarity reduction of soliton equations E.g. Modified KdV equation Painlevé II
mKdV eqn.
mKdV hierarchy Modified KP hierarchy
Painlevé II :
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Noumi-Yamada (1998)
Lie algebra Soliton eqs. → Painlevé eqs.
mKdV → Panlevé II
mBoussinesq → Panlevé IV
3-reduced KP → Panlevé V
・・・ ・・・ ・・・
n-reduced KP →Higher-order eq
s.
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Aim of this research
Consider the “multi-component” cases.
Multi-component KP hierarchy= KP hierarchy with matrix-coefficients
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From mKP hierarchy to Painlevé eqs.
mKP reduction Soliton eqs. Painlevé eqs.
1-component
2-reduced mKdV P II
3-reduced mBoussinesq P IV
4-reduced 4-reduced KP P V
n-reduced n-reduced KP Higher-order eqs.[Noumi-Yamada]
2-component(1,1) NLS P IV [Jimbo-Miwa]
(2,1) Yajima-Oikawa P V [Kikuchi-Ikeda-K]
3-component(1,1,1) 3-wave system P VI [K-Kikuchi]
… …
…
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Relation to affine Lie algebras
realization mKP soliton Painlevé
Principal 1-component, 2-reduced mKdV P II
Homogeneous 2-component, (1,1)-reduced NLS P IV
Principal 1-component, 3-reduced
mBoussinesq P IV
(2,1)-graded 2-component, (2,1)-reduced
Yajima-Oikawa P V
Homogeneous 3-component, (1,1,1)-reduced 3-wave P VI
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Rational solutions of Painlevé IVSchur polynomials Rational sol’s of P IV
1-component KP mBoussinesq P IV
“3-core” Okamoto polynomials[Kajiwara-Ohta], [Noumi-Yamada]
2-component KP derivative NLS P IV
“rectangular” Hermite polynomials[Kajiwara-Ohta], [K-Kikuchi]
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Aim of this research
Consider the multi-component cases.
Consider the meaning of similarity conditions at the level of the (modified) KP hierarchy.
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Multi-component mKP hierarchy Shift operator
Sato-Wilson operators
Sato equations
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1-component mKP hierarchy mKdV
2-reduction2-reduction
(modified KdV eq.)
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Proposition 1 Define as
where satisfies
Then also solve the Sato equations.
Scaling symmetry of mKP hierarchy
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1-component mKP mKdV P II
Similarity conditionSimilarity condition (mKdV P II)
2-reduction2-reduction (mKP mKdV)
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2-component mKP NLS P IV
Similarity condition Similarity condition (NLS P IV)
(1,1)-reduction(1,1)-reduction (2c-mKP NLS)
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Parameters in similarity conditions
Parameters in Painlevé equations
mKdV case (P II)
NLS case (P IV)
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Monodromy problemSimilarity condition Similarity condition (NLS P IV)
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Aim of this research
Consider the multi-component cases.
Consider the meaning of similarity conditions at the level of the (modified) KP hierarchy.
Consider the 3-component case to obtain the generic Painlevé VI.
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Three-wave interaction equations [Fokas-Yortsos], [Gromak-Tsegelnik], [Kitaev], [Duburovin-Mazzoco], [Conte-Grundland-Musette], [K-Kikuchi]
Self-dual Yang-Mills equation [Mason-Woodhouse], [Y. Murata], [Kawamuko-Nitta]
Schwarzian KdV Hierarchy [Nijhoff-Ramani-Grammaticos-Ohta], [Nijhoff-Hone-Joshi]
UC hierarchy [Tstuda], [Tsuda-Masuda] D4
(1)-type Drinfeld-Sokolov hierarchy [Fuji-Suzuki] Nonstandard 2 2 soliton system [M. Murata]
Painlevé VI as similarity reduction
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Painlevé VI as similarity reductionDirect approach based on three-wave system [Fokas-Yortsos (1986)] 3-wave PVI with 1-parameter
[Gromak-Tsegelnik (1989)] 3-wave PVI with 1-parameter
[Kitaev (1990)] 3-wave PVI with 2-parameters
[Conte-Grundland-Musette (2006)] 3-wave PVI with 4-parameters (arXiv:nlin.SI/0604011)
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Our approach (arXiv:nlin.SI/0508021)
3-component KP hierarchy
(1,1,1)-reduction
gl3-hierarhcy
Similarity reduction
3×3 monodromy problem
Laplace transformation
2×2 monodromy problem
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3-component KP 3-wave system
Compatibiliry
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3-wave system
3-component KP 3-wave system (1,1,1)-condition:
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3-component KP 3 3 system
Similarity conditionSimilarity condition
(1,1,1)-reduction(1,1,1)-reduction
cf. [Fokas-Yortsos]
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3-component KP 3 3 system Similarity conditionSimilarity condition
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3-component KP 3 3 system
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Laplace transformation with the condition :
3 3 2 2[Harnad, Dubrovin-Mazzocco, Boalch]
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Our approach (arXiv:nlin.SI/0508021)3-component KP hierarchy
(1,1,1)-reduction
gl3-hierarhcy
Similarity reduction
3×3 monodromy problem
Laplace transformation
2×2 monodromy problem P VI
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q-analogue (arXiv:nlin.SI/0605052)
3-component q-mKP hierarchy
(1,1,1)-reduction
q-gl3-hierarhcy
q-Similarity reduction
3×3 connection problem
q-Laplace transformation
2×2 connection problem q-P VI
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References SK, T. Kikuchi,
The sixth Painleve equation as similarity reduction of gl3 hierarchy, arXiv: nlin.SI/0508021
SK, T. Kikuchi, A q-analogue of gl3 hierarchy and q-Painleve VI, arXiv:nlin.SI/0605052
SK, T. Kikuchi,Affine Lie group approach to a derivative nonlinear Schrödinger equation and its similarity reduction,Int. Math. Res. Not. 78 (2004), 4181-4209
SK, T. Kikuchi,Solutions of a derivative nonlinear Schrödinger hierarchy and its similarity reduction,Glasgow Math. J. 47A (2005) 99-107
T. Kikuchi, T. Ikeda, SK, Similarity reduction of the modified Yajima-Oikawa equation,J. Phys. A36 (2003) 11465-11480