= Bright soliton of the focusing nonlinear Schrödinger equation
In the <normalization>
$$
i\psi_t=-\frac12\psi_{xx}-|\psi|^2\psi,
$$
the 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 $\kappa=\sqrt{2|E|}$. Then
$$
\psi(x,t)=e^{-iEt+i\theta_0}f_E(x-x_0),\qquad
f_E(x)=\kappa\operatorname{sech}(\kappa x).
$$
The 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 gives
$$
f_E''=\kappa^2f_E-2f_E^3.
$$
Hence $-f_E''/2-f_E^3=-\kappa^2f_E/2=Ef_E$, exactly the stationary equation. The peak amplitude is $\kappa$, the width scale is $1/\kappa$, and $\int_{\mathbb R}|f_E|^2dx=2\kappa$, so its <L2 norm> is finite.
A <Galilean boost of a nonlinear Schrödinger soliton> produces the moving solution
$$
\psi_u(x,t)=e^{i(ux-u^2t/2)}e^{-iEt+i\theta_0}f_E(x-ut-x_0).
$$
Its centre travels at constant <velocity> $u$, while its shape and <L2 norm> remain fixed. <Nonlinear dispersive equations: local and global analysis by Terence Tao>[Section 3.1, printed p. 114, equation 3.10: Galilean symmetry in the half-Laplacian normalization; the explicit cubic profile is verified above]{ref}
Back to article page
