Put $\mu=|E|=-E>0$ and seek $\phi=e^{-iEt}f(x)$ with real $f$. The equation becomes
$$
Ef=-\frac12f''-f^3.
$$
For $b=\sqrt{2\mu}$, the identity
$$
\frac{d^2}{dx^2}\operatorname{sech}(bx)
=b^2\operatorname{sech}(bx)
-2b^2\operatorname{sech}^3(bx)
$$
shows that
$$
\boxed{f_E(x)=\sqrt{2|E|}\operatorname{sech}(\sqrt{2|E|}x)}
$$
satisfies the stationary equation. Hence $e^{-iEt}f_E(x)$ is the <bright soliton of the focusing nonlinear Schrödinger equation>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 313 / 1 / i / Solution

For negative frequency $E$, the one-dimensional focusing nonlinear Schrödinger equation has the localized stationary profile
$$
f_E(x)=\sqrt{2|E|}\operatorname{sech}(\sqrt{2|E|}x).
$$
Galilean boosts generate moving profiles with a translational collective coordinate and constant velocity.


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