Travelling waves for nonlinear Schrödinger equation

Question

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 $c \in \mathbb{R}^n$.

Answer 1

For the case of Gross--Pitaevskii type nonlinearities, you might look at the Annals paper of Maris (as well as references therein and thereof). Here is an arXiv version: http://arxiv.org/abs/0903.0354

And, if you have access: http://annals.math.princeton.edu/wp-content/uploads/Maris.pdf