Article
Ufa Mathematical Journal
Volume 8, Number 4, pp. 52-61
On simultaneous solution of the KdV equation and a fifth-order differential equation
Garifullin R.N.
DOI:10.13108/2016-8-4-52
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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 < s < 5/24 $ 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 Painlevè IV equation.