Non-localized standing waves of the hyperbolic cubic nonlinear Schrödinger equation

We construct two families of non-localized standing waves for the hyperbolic cubic nonlinear Schrödinger equation\[iu_t+u_{xx}-u_{yy}+|u|^2u=0.\]The first family of standing waves consists of solutions which correspond to some generalized breathers for each fixed time $t$, while solutions in the second family are periodic both in $x$ and $y$. The second family of solutions were numerically observed by Vuillon, Dutykh and Fedele in a recent preprint [17].