Bright soliton of the focusing nonlinear Schrödinger equation

For negative frequency $E$, the one-dimensional focusing nonlinear Schrödinger equation has the localized stationary profileGalilean boosts generate moving profiles with a translational collective coordinate and constant velocity.

Let a real localized profile satisfy $Ef=-f^{\prime \prime }/2-f^3+y^{2}f/2$. Then $\psi (x,t)=e^{i\Theta (x,t)}f(x-X(t))$ solves the cubic focusing nonlinear Schrodinger equation in the potential $x^2/2$ with the displayed phase and any harmonic-oscillator trajectory $X$. Imaginary terms require ${\Theta }_x=\dot{X}$; real terms require $\ddot{X}=-X$ and the remaining time-dependent phase. The translation is exact and causes no profile deformation, unlike a generic collective-coordinate effective Lagrangian approximation.

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.