On simultaneous solution of the KdV equation and a fifth-order differential equation

Abstract

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 → ∞. For large time, the asymptotic solution has different structure depending on the slow variable s = x2/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 Painlevé IV equation.