Geodesic
On simultaneous solution of the KdV equation and a fifth-order differential equation
Ufa mathematical journal, Tome 8 (2016) no. 4, pp. 52-61 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the paper we consider an universal solution to the KdV equation. This solution also satisfies a fifth order ordinary differential equation. We pose the problem on studying the behavior of this solution as $t\to\infty$. For large time, the asymptotic solution has different structure depending on the slow variable $s=x^2/t$. We construct the asymptotic solution in the domains $s-3/4$, $-3/4$ and in the vicinity of the point $s=-3/4$. It is shown that a slow modulation of solution's parameters in the vicinity of the point $s=-3/4 $ is described by a solution to Painlevé IV equation.
Keywords: asymptotics, matching of asymptotic expansions, Korteweg–de Vries equation, non-dissipative shock waves. 
@article{UFA_2016_8_4_a3,
     author = {R. N. Garifullin},
     title = {On simultaneous solution of the {KdV} equation and a~fifth-order differential equation},
     journal = {Ufa mathematical journal},
     pages = {52--61},
     year = {2016},
     volume = {8},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2016_8_4_a3/}
}
TY  - JOUR
AU  - R. N. Garifullin
TI  - On simultaneous solution of the KdV equation and a fifth-order differential equation
JO  - Ufa mathematical journal
PY  - 2016
SP  - 52
EP  - 61
VL  - 8
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/UFA_2016_8_4_a3/
LA  - en
ID  - UFA_2016_8_4_a3
ER  - 
%0 Journal Article
%A R. N. Garifullin
%T On simultaneous solution of the KdV equation and a fifth-order differential equation
%J Ufa mathematical journal
%D 2016
%P 52-61
%V 8
%N 4
%U http://geodesic.mathdoc.fr/item/UFA_2016_8_4_a3/
%G en
%F UFA_2016_8_4_a3
R. N. Garifullin. On simultaneous solution of the KdV equation and a fifth-order differential equation. Ufa mathematical journal, Tome 8 (2016) no. 4, pp. 52-61. http://geodesic.mathdoc.fr/item/UFA_2016_8_4_a3/

[1] A. M. Il'in, S. V. Zakharov, “On the influence of small dissipation on the evolution of weak discontinuities”, International Conference on Differential and Functional Differential Equations (Moscow, 1999), Funct. Differ. Equ., 8:3–4 (2001), 257–271 | MR | Zbl

[2] Zakharov S. V., Ilin A. M., “Ot slabogo razryva k gradientnoi katastrofe”, Matem. sb., 192:10 (2001), 3–18 | DOI | MR | Zbl

[3] Zakharov S. V., “Zarozhdenie udarnoi volny v odnoi zadache Koshi dlya uravneniya Byurgersa”, Zh. vychisl. matem. i matem. fiz., 44:3 (2004), 536–542 | MR | Zbl

[4] Garifullin R. N., Suleimanov B. I., “Ot slabykh razryvov k bezdissipativnym udarnym volnam”, ZhETF, 137:1 (2010), 149–164

[5] Kamchatnov A. M., Korneev S. V., “Techenie Boze-Einshteinovskogo kondensata v kvaziodnomernom kanale pod deistviem porshnya”, ZhETF, 137:1 (2010), 191–204

[6] Ilin A. M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989, 336 pp.

[7] Garifullin R. N., “Sdvig fazy dlya sovmestnogo resheniya uravneniya KDV i differentsialnogo uravneniya pyatogo poryadka”, Ufimsk. matem. zhurn., 4:2 (2012), 80–86 | MR

[8] R. Garifullin, B. Suleimanov, N. Tarkhanov, “Phase Shift in the Whitham Zone for the Gurevich-Pitaevskii Special Solution of the Korteweg–de Vries Equation”, Ph. Let. A, 374 (2010), 1420–1424 | DOI | MR | Zbl

[9] Its A. R., Kapaev A. A., “Metod izomonodromnykh deformatsii i formuly svyazi dlya vtorogo transtsendenta Penleve”, Izv. AN SSSR. Ser. matem., 51:4 (1987), 878–892 | MR | Zbl