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From the KP hierarchy to the Painlevé equations

Painlev é 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

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From the KP hierarchy to the Painlevé equations

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

  2. 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 Sato’s KP hierarchy.

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

  4. Noumi-Yamada (1998)

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

  6. From mKP hierarchy to Painlevé eqs.

  7. Relation to affine Lie algebras

  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 1Define 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 Painlevé equations Parameters in similarity conditions • 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 2 2 soliton system [M. Murata]

  19. 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)

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

  24. 3-component KP 3 3 system Similarity condition

  25. 3-component KP 3 3 system

  26. 3 3 2 2 [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/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

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