 Cylindrical KdV equation

The cylindrical KdV equation in this normalization is $q_t+q_{xxx}+q{q}_x+q/(3t)=0$, for $t\ne 0$. The additional time-dependent term distinguishes it from the ordinary KdV equation. A scaling-invariant solution has $q=t^{-2/3}F(x{t}^{-1/3})$. Substitution gives $F^{(3)}+F{F}^{\prime }-(\xi F{)}^{\prime }/3=0$. Integration reduces the similarity problem to $F^{\prime \prime }+F^2/2-\xi F/3=C$. The integration constant depends on supplementary conditions and cannot be set to zero without them. Use real cube roots separately on either sign of $t$.