 Gauge reduction of a third-order half-line equation

For $Q_T+Q_{XXX}+a{Q}_{XX}+b{Q}_X+cQ=0$, an exponential gauge removes the second derivative. With ${\kappa }^2=a^2/3-b>0$ and $\sigma =-c+ab/3-2a^3/27$, the remaining equation is $q_t+q_{xxx}-q_x=0$. Negative $a$ improves spatial decay of the transformed initial data; positive $a$ can destroy the convergence of its Half-range Fourier transform.