Galilean boost of a nonlinear Schrödinger soliton (${\psi }_u(x,t)=e^{i(ux-u^{2}t/2)}{\psi }_0(x-ut,t)$)

For $i{\psi }_t=-{\psi }_{xx}/2-|\psi {|}^2\psi$, translation of a solution at velocity $u$ must be accompanied by the displayed phase. Direct differentiation cancels the terms proportional to ${\partial }_x{\psi }_0$, leaving the original equation. A standing profile of frequency $E=-{\kappa }^2/2$ thus gives $\kappa \mathrm{sech}(\kappa (x-ut-X_0))e^{iux-i(u^2/2+E)t+i{\theta }_0}$. Its centre travels uniformly, while its L2 norm is unchanged.