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

摘要

In this paper we consider the Gross-Pitaevskii equation iu t = Δu + u(1 − |u|2), where u is a complex-valued function defined on ${Bbb R}^Ntimes{Bbb R}$ , N ≥ 2, and in particular the travelling waves, i.e., the solutions of the form u(x, t) = ν(x 1 − ct, x 2, …, x N ), where $cin{Bbb R}$ 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.