Geodesic
Power and exponential expansions of solutions for the equation of the $K_{II}$ hierarchy
Trudy Instituta matematiki, Tome 16 (2008) no. 2, pp. 97-104 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Consider fourth order ordinary differential equation, which is a particular case of the $K_{II}$ hierarchy when $n=1$. We applied the Newton polygons method, evolved by A. D. Bruno. We used that method to find power, exponential expansions and their exponential additions for solutions of the equation. As the result we received a four-parametric family of holomorphic expansions, four asymptotic expansions, four families of polar expansions, holomorphic expansions around infinity and their exponential additions. When $\beta=0$ we had the particular cases of the expansions listed above, noted that for the holomorphic near infinity expansions the particular case were four exponential expansions. 
@article{TIMB_2008_16_2_a9,
     author = {M. S. Nialepka},
     title = {Power and exponential expansions of solutions for the equation of the $K_{II}$ hierarchy},
     journal = {Trudy Instituta matematiki},
     pages = {97--104},
     year = {2008},
     volume = {16},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMB_2008_16_2_a9/}
}
TY  - JOUR
AU  - M. S. Nialepka
TI  - Power and exponential expansions of solutions for the equation of the $K_{II}$ hierarchy
JO  - Trudy Instituta matematiki
PY  - 2008
SP  - 97
EP  - 104
VL  - 16
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TIMB_2008_16_2_a9/
LA  - ru
ID  - TIMB_2008_16_2_a9
ER  - 
%0 Journal Article
%A M. S. Nialepka
%T Power and exponential expansions of solutions for the equation of the $K_{II}$ hierarchy
%J Trudy Instituta matematiki
%D 2008
%P 97-104
%V 16
%N 2
%U http://geodesic.mathdoc.fr/item/TIMB_2008_16_2_a9/
%G ru
%F TIMB_2008_16_2_a9
M. S. Nialepka. Power and exponential expansions of solutions for the equation of the $K_{II}$ hierarchy. Trudy Instituta matematiki, Tome 16 (2008) no. 2, pp. 97-104. http://geodesic.mathdoc.fr/item/TIMB_2008_16_2_a9/

[1] Kudryashov N.A., “Transcendents defined by nonlinear fourth-order ordinary differential equations”, J. Phys. A: Math. Gen., 32 (1999), 999–1013 | DOI | MR | Zbl

[2] Kudryashov N.A., “Double Backlund transformations and special integrals for the $K_{II}$ hierarchy”, Phys. Lett. A, 273 (2000), 194–202 | DOI | MR | Zbl

[3] Bryuno A.D., Stepennaya geometriya v algebraicheskikh i differentsialnykh uravneniyakh, Fizmatlit, M., 1998 | MR | Zbl

[4] Bryuno A.D., “Asimptotiki i razlozheniya reshenii obyknovennogo differentsialnogo uravneniya”, Uspekhi matematicheskikh nauk, 59:3 (2004), 31–80 | MR | Zbl

[5] Bryuno A.D., Zavgordnyaya Yu.V., Stepennye ryady i nestepennye asimptotiki reshenii vtorogo uravneniya Penleve, Preprint No 48, In-t prikladnoi matematiki im. M.V. Keldysha RAN, M., 2003

[6] Wasow V.R., Asymptotic expansions for the ordinary differential equations, John Wiley and Sons, New-York; London; Sydney, 1965, 75 pp. | MR | Zbl