How to show the nonlinear Schrodinger equation is $S^1$-symmetry?

Question

In Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials, the author say the nonlinear Schrodinger equation $$ ih\frac{\partial \psi}{\partial t} = -\frac{h^2}{2}\Delta \psi + V\psi-|\psi|^{p-1}\psi $$ is $S^1$-symmetry. How to show it ? In fact, I don't know how to structure the transformation of $S^1$-symmetry.

Answer 1

In this case we view the group $S^1$ as the unit circle in the complex plane with the group law being complex multiplication. The symmetry in question is not very sophisticated. It simply says that if $\psi$ solves $$ ih\frac{\partial \psi}{\partial t} = -\frac{h^2}{2}\Delta \psi + V\psi-|\psi|^{p-1}\psi $$ and $e^{i \omega} \in S^1$, then the new function $\phi = e^{i \omega} \psi$ solves $$ ih\frac{\partial \phi}{\partial t} = -\frac{h^2}{2}\Delta \phi + V\phi-|\phi|^{p-1}\phi. $$ This follows by multiplying the PDE for $\psi$ by the complex constant $e^{i \omega}$ and noting that $|\psi| = |e^{i\omega} \psi| = |\phi|$.