1. From the KP hierarchy to the Painlevé equations
Painlevé Equations and Monodromy Problems: Recent Developments
From the KP hierarchy to the Painlevé equations
Saburo KAKEI (Rikkyo University)
Joint work with Tetsuya KIKUCHI (University of Tokyo)
22 September 2006

2. Known Facts
Fact 1
Painlevé equations can be obtained as similarity reduction of soliton equations.
Fact 2 Many (pahaps all) soliton equations can be obtained as reduced cases of Sato’s KP hierarchy.

3. Similarity reduction of soliton equations
E.g. Modified KdV equation Painlevé II
mKdV hierarchy
Modified KP hierarchy
mKdV eqn.
Painlevé II：
Similarity

4. 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.

5. Aim of this research
Consider the “multi-component” cases. Multi-component KP hierarchy = KP hierarchy with matrix-coefficients

6. Higher-order eqs. [Noumi-Yamada]
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] …

7. 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
1-component, 3-reduced
mBoussinesq
(2,1)-graded
2-component, (2,1)-reduced
Yajima-Oikawa
P V
3-component, (1,1,1)-reduced
3-wave
P VI

8. Rational solutions of Painlevé IV
Schur 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]

9. Aim of this research Consider the multi-component cases.
Consider the meaning of similarity conditions at the level of the (modified) KP hierarchy.

10. Multi-component mKP hierarchy
Shift operator
Sato-Wilson operators
Sato equations

11. 1-component mKP hierarchy mKdV
2-reduction
(modified KdV eq.)

12. Scaling symmetry of mKP hierarchy
Proposition 1　Define as
　where satisfies
Then also solve the Sato equations.

13. 1-component mKP mKdV P II
2-reduction (mKP mKdV)
Similarity condition (mKdV P II)

14. 2-component mKP NLS P IV (1,1)-reduction (2c-mKP NLS)
Similarity condition (NLS P IV)

15. Parameters in similarity conditions Parameters in Painlevé equations
mKdV case (P II)
NLS case (P IV)

16. Monodromy problem
Similarity condition (NLS P IV)

17. 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.

18. 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-Grammaticos-Ohta], [Nijhoff-Hone-Joshi]
UC hierarchy [Tstuda], [Tsuda-Masuda]
D4(1)-type Drinfeld-Sokolov hierarchy [Fuji-Suzuki]
Nonstandard soliton system [M. Murata]

19. Painlevé VI as similarity reduction
Direct approach based on three-wave system
[Fokas-Yortsos (1986)] wave PVI with 1-parameter
[Gromak-Tsegelnik (1989)] 3-wave PVI with 1-parameter
[Kitaev (1990)] wave PVI with 2-parameters
[Conte-Grundland-Musette (2006)] 3-wave PVI with 4-parameters (arXiv:nlin.SI/ )

20. 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

21. 3-component KP 3-wave system
Compatibiliry

22. 3-component KP 3-wave system
(1,1,1)-condition:
3-wave system

23. 3-component KP 3 3 system (1,1,1)-reduction Similarity condition
cf. [Fokas-Yortsos]

24. 3-component KP system
Similarity condition

25. 3-component KP system

26. 3 3 2 2 [Harnad, Dubrovin-Mazzocco, Boalch]
[Harnad, Dubrovin-Mazzocco, Boalch]
Laplace transformation with the condition :

27. 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

28. 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

29. References
SK, T. Kikuchi, The sixth Painleve equation as similarity reduction of gl3 hierarchy, arXiv: nlin.SI/
SK, T. Kikuchi, A q-analogue of gl3 hierarchy and q-Painleve VI, arXiv:nlin.SI/
SK, T. Kikuchi, Affine Lie group approach to a derivative nonlinear Schrödinger equation and its similarity reduction, Int. Math. Res. Not. 78 (2004),
SK, T. Kikuchi, Solutions of a derivative nonlinear Schrödinger hierarchy and its similarity reduction, Glasgow Math. J. 47A (2005)
T. Kikuchi, T. Ikeda, SK, Similarity reduction of the modified Yajima-Oikawa equation, J. Phys. A36 (2003)