Geodesic
A method for the numerical solution of the Painlevé equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 5, pp. 702-726 
A. A. Abramov; L. F. Yukhno. A method for the numerical solution of the Painlevé equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 5, pp. 702-726. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_5_a2/
@article{ZVMMF_2013_53_5_a2,
     author = {A. A. Abramov and L. F. Yukhno},
     title = {A method for the numerical solution of the {Painlev\'e} equations},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {702--726},
     year = {2013},
     volume = {53},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_5_a2/}
}
TY  - JOUR
AU  - A. A. Abramov
AU  - L. F. Yukhno
TI  - A method for the numerical solution of the Painlevé equations
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2013
SP  - 702
EP  - 726
VL  - 53
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_5_a2/
LA  - ru
ID  - ZVMMF_2013_53_5_a2
ER  - 
%0 Journal Article
%A A. A. Abramov
%A L. F. Yukhno
%T A method for the numerical solution of the Painlevé equations
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2013
%P 702-726
%V 53
%N 5
%U http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_5_a2/
%G ru
%F ZVMMF_2013_53_5_a2

Voir la notice de l'article provenant de la source Math-Net.Ru

A numerical method for solving the Cauchy problem for all the six Painlevé equations is proposed. The difficulty of solving these equations is that the unknown functions can have movable (that is, dependent on the initial data) singular points of the pole type. Moreover, the Painlevé III–VI equations may have singularities at points where the solution takes certain finite values. The positions of all these singularities are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities at the corresponding point and its neighborhood. Such auxiliary equations are derived for all Painlevé equations and for all types of singularities. Efficient criteria for transition to auxiliary systems are formulated, and numerical results illustrating the potentials of the method are presented.

[1] Abramov A. A., Yukhno L. F., “Chislennoe reshenie zadachi Koshi dlya uravnenii Penleve I, II”, Zh. vychisl. matem. i matem. fiz., 52:3 (2012), 379–387 | Zbl

[2] Abramov A. A., Yukhno L. F., “Chislennoe reshenie zadachi Koshi dlya uravneniya Penleve III”, Differents. ur-niya, 48:7 (2012), 925–934 | MR | Zbl

[3] Abramov A. A., Yukhno L. F., “Chislennoe reshenie uravneniya Penleve IV”, Zh. vychisl. matem. i matem. fiz., 52:11 (2012), 2023–2032 | MR | Zbl

[4] Abramov A. A., Yukhno L. F., “Chislennoe reshenie uravneniya Penleve V”, Zh. vychisl. matem. i matem. fiz., 53:1 (2013), 58–71 | DOI | MR | Zbl

[5] Abramov A. A., Yukhno L. F., “Chislennoe reshenie uravneniya Penleve VI”, Zh. vychisl. matem. i matem. fiz., 53:2 (2013), 249–262 | DOI | Zbl

[6] Its A. R., Kapaev A. A., Novokshenov V. Yu., Fokas A. S., Transtsendenty Penleve. Metod zadachi Rimana, Institut kompyuternykh issledovanii, NITs “Regulyarnaya i khaoticheskaya dinamika”, M.–Izhevsk, 2005 | Zbl

[7] Erugin N. P., “Teoriya podvizhnykh osobykh tochek dlya uravnenii vtorogo poryadka, I”, Differents. ur-niya, 12:3 (1976), 387–416 | MR | Zbl

[8] Erugin N. P., “Teoriya podvizhnykh osobykh tochek dlya uravnenii vtorogo poryadka, II”, Differents. ur-niya, 12:4 (1976), 579–598 | MR | Zbl