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 $t\to\infty$. 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