Non-existence for travelling waves with small energy for the Gross–Pitaevskii equation in dimension larger than or equal to three

We prove that the Ginzburg-Landau energy of non-constant travelling waves of the Gross-Pitaevskii equation has a lower positive bound, depending only on the dimension, in any dimension larger or equal to three. In particular, we conlude that there are no non-constant travelling waves with small energy