Global dispersive solutions for the Gross-Pitaevskii equation in two and   three dimensions
  We study asymptotic behaviour at time infinity of solutions close to the non-zero constant equilibrium for the Gross-Pitaevskii equation in two and three spatial dimensions. We construct a class of global solutions with prescribed dispersive asymptotic behavior, which is given in terms of the linearized evolution. 
S. Gustafson, K. Nakanishi, T.-P. Tsai