Abstract.
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 \({\Bbb R}^N\times{\Bbb R}\), 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 \(c\in{\Bbb R}\) 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.
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Tarquini, E. A lower bound on the energy of travelling waves of fixed speed for the Gross-Pitaevskii equation. Mh Math 151, 333–339 (2007). https://doi.org/10.1007/s00605-006-0443-3
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DOI: https://doi.org/10.1007/s00605-006-0443-3