# Travelling waves for nonlinear Schrödinger equation

Consider the following nonlinear Schrödinger equation: $$-\Delta \Phi - i\frac{\partial \Phi}{\partial t} = f(|\Phi|^2)\Phi,$$ where $\Delta$ is the Laplacian on $\mathbb{R}^n$, $f$ gives the nonlinearity (I am not specifying any conditions on $f$, since I want to make this a broad reference request; $f$ can be power-type or Gross-Pitaevskii type, or otherwise). I am looking for references for existence/non-existence of (super/sub)sonic travelling wave solutions of the above equation with null condition at infinity. By travelling wave solution, I mean a solution of the form $\Phi(x, t) = \varphi (x + ct)$, where $c \in \mathbb{R}^n$.