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

Abstract

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.