On simultaneous solution of the KdV equation and a~fifth-order differential equation
Ufa mathematical journal, Tome 8 (2016) no. 4, pp. 52-61
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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},
publisher = {mathdoc},
volume = {8},
number = {4},
year = {2016},
language = {en},
url = {http://geodesic.mathdoc.fr/item/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/