Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 1 / 31E / iii / Solution

Balancing the time derivative, cubic nonlinearity and third spatial derivative gives $x\sim t^{1/3}$ and $u\sim t^{-1/3}$. Choose $\zeta =x/(3t{)}^{1/3}$ and $u=(3t{)}^{-1/3}w(\zeta )$. ThenThus the similarity solution isThe extra spherical term changes the coefficient of $w$. This third-order equation cannot be integrated directly into the standard second-order Painlevé II equation of the ordinary modified KdV reduction.