= Cylindrical KdV equation
{title2=$q_t+q_{xxx}+qq_x+q/(3t)=0$}
The <cylindrical KdV equation> in this normalization is $q_t+q_{xxx}+qq_x+q/(3t)=0$, for $t\ne0$. The additional time-dependent term distinguishes it from the ordinary <KdV equation>. A scaling-invariant solution has $q=t^{-2/3}F(xt^{-1/3})$. Substitution gives $F^{(3)}+FF'-(\xi F)'/3=0$. <Integration> reduces the similarity problem to $F''+F^2/2-\xi F/3=C$. The <integration> constant depends on supplementary conditions and cannot be set to zero without them. Use real cube roots separately on either sign of $t$.
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