Vortex rings for the Gross-Pitaevskii equation in \(\mathbb R^3\) (Q385980)

The authors consider the Gross-Pitaevskii equation NEWLINE\[NEWLINEiu_t=\varepsilon^2\Delta u+(V-| u|^2)u NEWLINE\]NEWLINE in \(\mathbb R^{1+3}\). It is proved that for any small \(\varepsilon>0\) this equation has a radially symmetric stationary solution with a vortex ring of degree one. Existence of traveling wave solution with a traveling vortex ring of degree one is also proved for small \(\varepsilon>0\).