Weights, Kovalevskaya exponents and the Painlevé property
[Poids, exposants de Kovalevskaya et propriété de Painlevé]
Annales de l'Institut Fourier, Online first, 38 p.

On étudie les degrés pondérés des fonctions hamiltoniennes quasi homogènes des équations de Painlevé. On classifie un t-uplet d’entiers positifs, appelé poids régulier, satisfaisant certaines conditions liées à la théorie de la singularité. A chaque équation polynomiale de Painlevé est associé un poids régulier. Inversement, pour les cas des dimensions 2 et 4, on montre qu’il existe une équation différentielle satisfaisant la propriété de Painlevé associée à chaque poids régulier. Les exposants de Kovalevskaya des systèmes hamiltoniens quasi homogènes sont également étudiés au moyen de poids réguliers, la théorie des singularités et la théorie des systèmes dynamiques. On montre qu’il existe une correspondance biunivoque entre les solutions de la série de Laurent et des variétés stables du champ vectoriel associé obtenues par l’éclatement du système. Pour des équations autonomes de Painlevé en dimension 4, la surface de niveau des fonctions hamiltoniennes peut être décomposée en une union disjointe de variétés stables.

Weighted degrees of quasihomogeneous Hamiltonian functions of the Painlevé equations are investigated. A t-uple of positive integers, called a regular weight, satisfying certain conditions related to singularity theory is classified. For each polynomial Painlevé equation a regular weight is associated. Conversely, for 2 and 4-dim cases, it is shown that there exists a differential equation satisfying the Painlevé property associated with each regular weight. Kovalevskaya exponents of quasihomogeneous Hamiltonian systems are also investigated by means of regular weights, singularity theory and dynamical systems theory. It is shown that there is a one-to-one correspondence between Laurent series solutions and stable manifolds of the associated vector field obtained by the blow-up of the system. For 4-dim autonomous Painlevé equations, the level surface of Hamiltonian functions can be decomposed into a disjoint union of stable manifolds.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3591
Classification : 34M55
Keywords: Painlevé equation, quasihomogeneous vector field, regular weight, Kovalevskaya exponent
Mot clés : Équation de Painlevé, champ vectoriel quasi homogéne, poids normal, exposant de Kovalevskaya
Chiba, Hayato 1

1 Advanced Institute for Materials Research Tohoku University Sendai, 980-8577 (Japan)
@unpublished{AIF_0__0_0_A51_0,
     author = {Chiba, Hayato},
     title = {Weights, {Kovalevskaya} exponents and the {Painlev\'e} property},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2023},
     doi = {10.5802/aif.3591},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Chiba, Hayato
TI  - Weights, Kovalevskaya exponents and the Painlevé property
JO  - Annales de l'Institut Fourier
PY  - 2023
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3591
LA  - en
ID  - AIF_0__0_0_A51_0
ER  - 
%0 Unpublished Work
%A Chiba, Hayato
%T Weights, Kovalevskaya exponents and the Painlevé property
%J Annales de l'Institut Fourier
%D 2023
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3591
%G en
%F AIF_0__0_0_A51_0
Chiba, Hayato. Weights, Kovalevskaya exponents and the Painlevé property. Annales de l'Institut Fourier, Online first, 38 p.

[1] Ablowitz, M. J.; Ramani, A.; Segur, H. A connection between nonlinear evolution equations and ordinary differential equations of P-type. I, J. Math. Phys., Volume 21 (1980) no. 4, pp. 715-721 | DOI | MR | Zbl

[2] Adler, Mark; van Moerbeke, Pierre; Vanhaecke, Pol Algebraic integrability, Painlevé geometry and Lie algebras, Springer-Verlag, Berlin, 2004, xii+483 pages | DOI | MR

[3] Arnold, V. I.; Goryunov, V. V.; Lyashko, O. V.; Vasil’ev, V. A. Singularity theory. I, Springer-Verlag, Berlin, 1998, iv+245 pages | DOI | MR | Zbl

[4] Borisov, A. V.; Dudoladov, S. L. Kovalevskaya exponents and Poisson structures, Regul. Chaotic Dyn., Volume 4 (1999) no. 3, pp. 13-20 | DOI | MR | Zbl

[5] Chiba, Hayato Kovalevskaya exponents and the space of initial conditions of a quasi-homogeneous vector field, J. Differential Equations, Volume 259 (2015) no. 12, pp. 7681-7716 | DOI | MR | Zbl

[6] Chiba, Hayato The first, second and fourth Painlevé equations on weighted projective spaces, J. Differential Equations, Volume 260 (2016) no. 2, pp. 1263-1313 | DOI | MR | Zbl

[7] Chiba, Hayato The third, fifth and sixth Painlevé equations on weighted projective spaces, SIGMA Symmetry Integrability Geom. Methods Appl., Volume 12 (2016), p. 019 | DOI | MR | Zbl

[8] Chiba, Hayato Multi-Poisson approach to the Painlevé equations: from the isospectral deformation to the isomonodromic deformation, SIGMA Symmetry Integrability Geom. Methods Appl., Volume 13 (2017), p. 025 | DOI | MR | Zbl

[9] Clarkson, Peter A.; Joshi, Nalini; Mazzocco, Marta The Lax pair for the mKdV hierarchy, Théories asymptotiques et équations de Painlevé (Sémin. Congr.), Volume 14, Soc. Math. France, Paris, 2006, pp. 53-64 | MR | Zbl

[10] Clarkson, Peter A.; Joshi, Nalini; Pickering, Andrew Bäcklund transformations for the second Painlevé hierarchy: a modified truncation approach, Inverse Problems, Volume 15 (1999) no. 1, pp. 175-187 | DOI | MR | Zbl

[11] Cosgrove, Christopher M. Chazy classes IX–XI of third-order differential equations, Stud. Appl. Math., Volume 104 (2000) no. 3, pp. 171-228 | DOI | MR | Zbl

[12] Drinfel’d, V. G.; Sokolov, V. V. Lie algebras and equations of Korteweg–de Vries type, Current problems in mathematics, Vol. 24, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 81-180 | MR | Zbl

[13] Ercolani, Nicholas; Siggia, Eric D. Painlevé property and geometry, Phys. D, Volume 34 (1989) no. 3, pp. 303-346 | DOI | MR | Zbl

[14] Fuji, Kenta; Suzuki, Takao Drinfeld–Sokolov hierarchies of type A and fourth order Painlevé systems, Funkcial. Ekvac., Volume 53 (2010) no. 1, pp. 143-167 | DOI | MR | Zbl

[15] Gordoa, Pilar R.; Joshi, Nalini; Pickering, Andrew On a generalized 2+1 dispersive water wave hierarchy, Publ. Res. Inst. Math. Sci., Volume 37 (2001) no. 3, pp. 327-347 http://projecteuclid.org/euclid.prims/1145477227 | DOI | MR | Zbl

[16] Goriely, Alain Integrability and nonintegrability of dynamical systems, Advanced Series in Nonlinear Dynamics, 19, World Scientific Publishing Co., Inc., River Edge, NJ, 2001, xviii+415 pages | DOI | MR | Zbl

[17] Goriely, Alain Painlevé analysis and normal forms theory, Phys. D, Volume 152/153 (2001), pp. 124-144 | DOI | MR | Zbl

[18] de Groot, Mark F.; Hollowood, Timothy J.; Miramontes, J. Luis Generalized Drinfel’d–Sokolov hierarchies, Comm. Math. Phys., Volume 145 (1992) no. 1, pp. 57-84 http://projecteuclid.org/euclid.cmp/1104249535 | DOI | MR | Zbl

[19] Hu, J.; Yan, M. Painlevé Test and the Resolution of Singularities for Integrable Equations (https://arxiv.org/abs/1304.7982)

[20] Kawakami, Hiroshi Four-dimensional Painlevé-type equations associated with ramified linear equations I: Matrix Painlevé systems, Funkcial. Ekvac., Volume 63 (2020) no. 1, pp. 97-132 | DOI | MR | Zbl

[21] Kawakami, Hiroshi; Nakamura, Akane; Sakai, Hidetaka Degeneration scheme of 4-dimensional Painlevé-type equations, 4-dimensional Painlevé-type equations (MSJ Mem.), Volume 37, Math. Soc. Japan, Tokyo, 2018, pp. 25-111 | DOI | MR

[22] Koike, Tatsuya On new expressions of the Painlevé hierarchies, Algebraic analysis and the exact WKB analysis for systems of differential equations (RIMS Kôkyûroku Bessatsu, B5), Res. Inst. Math. Sci. (RIMS), Kyoto, 2008, pp. 153-198 | MR | Zbl

[23] Kudryashov, Nicolai A. The first and second Painlevé equations of higher order and some relations between them, Phys. Lett. A, Volume 224 (1997) no. 6, pp. 353-360 | DOI | MR | Zbl

[24] Matano, Tohru; Matumiya, Atusi; Takano, Kyoichi On some Hamiltonian structures of Painlevé systems. II, J. Math. Soc. Japan, Volume 51 (1999) no. 4, pp. 843-866 | DOI | MR | Zbl

[25] Matumiya, Atusi On some Hamiltonian structures of Painlevé systems. III, Kumamoto J. Math., Volume 10 (1997), pp. 45-73 | MR | Zbl

[26] Nakamura, Akane Autonomous limit of the 4-dimensional Painlevé-type equations and degeneration of curves of genus two, Ann. Inst. Fourier, Volume 69 (2019) no. 2, pp. 845-893 http://aif.cedram.org/item?id=AIF_2019__69_2_845_0 | DOI | MR | Zbl

[27] Noumi, Masatoshi; Yamada, Yasuhiko Higher order Painlevé equations of type A l (1) , Funkcial. Ekvac., Volume 41 (1998) no. 3, pp. 483-503 http://www.math.kobe-u.ac.jp/~fe/xml/mr1676885.xml | MR | Zbl

[28] Okamoto, Kazuo Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé, Japan. J. Math. (N.S.), Volume 5 (1979) no. 1, pp. 1-79 | DOI | MR | Zbl

[29] Saito, Kyoji Regular system of weights and associated singularities, Complex analytic singularities (Adv. Stud. Pure Math.), Volume 8, North-Holland, Amsterdam, 1987, pp. 479-526 | DOI | MR | Zbl

[30] Sakai, Hidetaka Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys., Volume 220 (2001) no. 1, pp. 165-229 | DOI | MR | Zbl

[31] Shimomura, Shun A certain expression of the first Painlevé hierarchy, Proc. Japan Acad. Ser. A Math. Sci., Volume 80 (2004) no. 6, pp. 105-109 http://projecteuclid.org/euclid.pja/1116014786 | DOI | MR | Zbl

[32] Shioda, Tsutomu; Takano, Kyoichi On some Hamiltonian structures of Painlevé systems. I, Funkcial. Ekvac., Volume 40 (1997) no. 2, pp. 271-291 http://www.math.kobe-u.ac.jp/~fe/xml/mr1480279.xml | MR | Zbl

[33] Yoshida, H.; Grammaticos, B.; Ramani, A. Painlevé resonances versus Kowalevski exponents: some exact results on singularity structure and integrability of dynamical systems, Acta Appl. Math., Volume 8 (1987) no. 1, pp. 75-103 | DOI | MR | Zbl

[34] Yoshida, Haruo Necessary condition for the existence of algebraic first integrals. I. Kowalevski’s exponents, Celestial Mech., Volume 31 (1983) no. 4, pp. 363-379 | DOI | MR | Zbl

Cité par Sources :