 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]