Phase shift for the common solution of the KdV and the fifth order differential equation

We investigate the special solution of Korteweg–de Vries equation. This solution describes the influence of small dispersion to a process of transformation from weak to strong discontinuities in inviscid fluid dynamics. This solution also satisfies the fifth order ordinary differential equation. We construct the asymptotic solution in the Witham zone up to a phase shift. We obtain an the equation for phase shift and, using the numerical experiments, we choose the concrete solution of this equation. This solution is a constant function.