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

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.

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

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.

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

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] …

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

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]

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

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

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

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

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

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

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

Monodromy problem Similarity condition (NLS P IV)

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.

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 2 2 soliton system [M. Murata]

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 (arXiv:nlin.SI/0604011)

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

3-component KP 3-wave system Compatibiliry

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

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

3-component KP 3 3 system Similarity condition

3-component KP 3 3 system

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

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

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

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