From the KP hierarchy to the Painlevé equations

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

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.

Similarity

Similarity reduction of soliton equations E.g. Modified KdV equation Painlevé II

mKdV eqn.

mKdV hierarchy Modified KP hierarchy

Painlevé II ：

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.

Aim of this research Consider the “multi-component” cases.

Multi-component KP hierarchy= KP hierarchy with matrix-coefficients

From mKP hierarchy to Painlevé eqs.

mKP reduction Soliton eqs. Painlevé eqs.

1-component

2-reduced mKdV P II3-reduced mBoussinesq P IV4-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

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

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]

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-reduction2-reduction

(modified KdV eq.)

Proposition 1　 Define as

　　 where satisfies

Then also solve the Sato equations.

Scaling symmetry of mKP hierarchy

1-component mKP mKdV P II

Similarity conditionSimilarity condition (mKdV P II)

2-reduction2-reduction (mKP mKdV)

2-component mKP NLS P IV

Similarity condition Similarity condition (NLS P IV)

(1,1)-reduction(1,1)-reduction (2c-mKP NLS)

Parameters in similarity conditions

Parameters in Painlevé equations

mKdV case (P II)

NLS case (P IV)

Monodromy problemSimilarity condition 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.

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

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)

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-wave system

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

3-component KP 3 3 system

Similarity conditionSimilarity condition

(1,1,1)-reduction(1,1,1)-reduction

cf. [Fokas-Yortsos]

3-component KP 3 3 system Similarity conditionSimilarity condition

3-component KP 3 3 system

Laplace transformation with the condition :

3 3 2 2[Harnad, Dubrovin-Mazzocco, Boalch]

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