From the KP hierarchy to the Painlev

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From the KP hierarchy to the Painlevé equations
Painlevé Equations and Monodromy Problems Recent
Developments

• Saburo KAKEI (Rikkyo University)
• Joint work with Tetsuya KIKUCHI (University of
  Tokyo)

22 September 2006
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Known Facts

• Fact 1
• Painlevé equations can be obtained as similarity
  reduction of soliton equations.
• Fact 2Many (pahaps all) soliton equations can be
  obtained as reduced cases of Satos KP hierarchy.

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Similarity reduction of soliton equations

• E.g. Modified KdV equation Painlevé II

mKdV hierarchy
Modified KP hierarchy
mKdV eqn.
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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 eqs.
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Aim of this research

• Consider the multi-component cases.Multi-comp
  onent 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
1-component 3-reduced mBoussinesq P IV
1-component 4-reduced 4-reduced KP P V
1-component n-reduced n-reduced KP Higher-order eqs.Noumi-Yamada
2-component (1,1) NLS P IV Jimbo-Miwa
2-component (2,1) Yajima-Oikawa P V Kikuchi-Ikeda-K
3-component (1,1,1) 3-wave system P VI K-Kikuchi
3-component
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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 IV
Schur polynomials Rational sols 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-reduction
(modified KdV eq.)
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Scaling symmetry of mKP hierarchy
Proposition 1 Define
as where satisfies Then
also solve the Sato
equations.
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1-component mKP mKdV P II
2-reduction (mKP mKdV)
Similarity condition (mKdV P II)
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2-component mKP NLS P IV
(1,1)-reduction (2c-mKP NLS)
Similarity condition (NLS P IV)
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Parameters in Painlevé equations
Parameters in similarity conditions

• mKdV case (P II)
• NLS case (P IV)

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Monodromy problem
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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Painlevé VI as similarity reduction

• 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-Grammati
  cos-Ohta, Nijhoff-Hone-Joshi
• UC hierarchy Tstuda, Tsuda-Masuda
• D4(1)-type Drinfeld-Sokolov hierarchy
  Fuji-Suzuki
• Nonstandard 2 2 soliton system M. Murata

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Painlevé VI as similarity reduction
Direct 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
(arXivnlin.SI/0604011)
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Our approach (arXivnlin.SI/0508021)

• 3-component KP hierarchy
• (1,1,1)-reduction
• gl3-hierarhcy
• Similarity reduction
• 33 monodromy problem
• Laplace transformation
• 22 monodromy problem

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3-component KP 3-wave system
Compatibiliry
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3-component KP 3-wave system
(1,1,1)-condition
3-wave system
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3-component KP 3 3 system
(1,1,1)-reduction
Similarity condition
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3-component KP 3 3 system
Similarity condition
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3-component KP 3 3 system
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3 3 2 2
Harnad, Dubrovin-Mazzocco, Boalch
Laplace transformation with the condition
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Our approach (arXivnlin.SI/0508021)

• 3-component KP hierarchy
• (1,1,1)-reduction
• gl3-hierarhcy
• Similarity reduction
• 33 monodromy problem
• Laplace transformation
• 22 monodromy problem

P VI
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q-analogue (arXivnlin.SI/0605052)
3-component q-mKP hierarchy
(1,1,1)-reduction q-gl3-hierarhcy
q-Similarity reduction 33 connection
problem q-Laplace transformation 22
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, arXivnlin.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