A lower bound on the energy of travelling waves of fixed speed for the Gross-Pitaevskii equation

In this paper we consider the Gross-Pitaevskii equation iu(t) = Delta u + u(1 - vertical bar u vertical bar(2)), where u is a complex- valued function defined on R-N x R, N >= 2, and in particular the travelling waves, i. e., the solutions of the form u(x, t) = v(x(1) - ct, x(2),..., x(N)), where c epsilon R is the speed. We prove for c fixed the existence of a lower bound on the energy of any non- constant travelling wave. This bound provides a non-existence result for non-constant travelling waves of fixed speed having small energy.