Bright soliton of the focusing nonlinear Schrödinger equation

In the normalizationthe one-dimensional focusing nonlinear Schrodinger equation admits a localized soliton on zero background. The negative cubic term balances the spreading caused by dispersion. For frequency $E<0$, let $\kappa =\sqrt{2|E|}$. ThenThe constants $x_0$ and ${\theta }_0$ specify the centre and constant phase. The coefficient $1/2$ of the second derivative fixes the relation between frequency, width and amplitude; other common nonlinear Schrödinger equation conventions rescale these quantities.

To check the solution, the hyperbolic secant identity givesHence $-f_E^{\prime \prime }/2-f_E^3=-{\kappa }^2{f}_E/2=E{f}_E$, exactly the stationary equation. The peak amplitude is $\kappa$, the width scale is $1/\kappa$, and ${\int }_{\mathbb{R}}|f_E{|}^{2}dx=2\kappa$, so its L2 norm is finite.

A Galilean boost of a nonlinear Schrödinger soliton produces the moving solutionIts centre travels at constant velocity $u$, while its shape and L2 norm remain fixed. ^[ref]

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.