= Galilean boost of a nonlinear Schrödinger soliton
{c}
{title2=$\psi_u(x,t)=e^{i(ux-u^2t/2)}\psi_0(x-ut,t)$}
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 $\kappa\operatorname{sech}(\kappa(x-ut-X_0))e^{iux-i(u^2/2+E)t+i\theta_0}$. Its centre travels uniformly, while its <L2 norm> is unchanged.
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