On the existence of certain elliptic solutions of the cubically nonlinear Schrödinger equation

We consider solutions of the cubically nonlinear Schrödinger equation. For a certain class of solutions of the form $\Psi(t,z)=(f(t,z)+id(z))e^{i\phi(z)}$ with $f,\phi,d\in\mathbb{R}$, we prove that they are nonexistent in the general case $f_z\neq 0$, $f_t\neq 0$, $d_z\neq 0$. In the three nongeneric cases ($f_z\neq 0$), ($f_t\neq 0$, $f_t=0$, $d_z=0$), and ($f_z=0$, $f_t\neq 0$), we present a two-parameter set of solutions, for which we find the constraints specifying real bounded and unbounded solutions.