Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 154 / 1 / 4 / Solution

Set $z=x-ct$ and $u(t,x)=Q_c(z)$. Substitution into the equation givesDecay at infinity makes the integration constant zero. Ifthen every term in $Q_c^{\prime \prime }-c{Q}_c+Q_c^p$ equals $c^{1+1/(p-1)}$ times the corresponding term in $Q^{\prime \prime }-Q+Q^p$. Thus the required Generalized KdV solitary wave is