Summary

In this paper, we developed a theory of weak inhomogeneous wave turbulence for BEC systems. We started with the GP equation and derived a statistical theory for the BEC kinetics which, in particular, describes states which are very far from the thermodynamic equilibrium. Such nonequilibrium states take the form of wave turbulence which is essentially inhomogeneous due to the fact that the BEC is trapped by an external field. There are two main new results in this paper. First of all, we have described the effect of the inhomogeneous ground state on the linear wave dynamics and, in particular, we have shown that such an effect cannot be modeled by renormalizing the trapping potential as it was previously suggested in literature. This was done by deriving a consistent WKB theory based on the scale separation between the ground state and the waves. Our results show that the condensate ``mildly'' pushes the wave turbulence away from the center but it can never reflect it (as an external potential would). Note that we established this result only for the limit of large occupation numbers described by the GP equation and this, in principle, does not rule out a possibility that the the renormalized potential approach can still be valid in the opposite limit of small occupation numbers. Secondly, we showed that the kinetic equation for trapped waves generalizes, and one can combine the linear WKB theory and the theory of homogeneous weak turbulence in a straightforward manner. Namely, the partial time derivative on the LHS of the kinetic equation is replaced by the full time derivative along the wave rays, while the frequency and the spectrum on the RHS now become functions of coordinate. A suitable definition for the coordinate dependent spectrum is given by using the Gabor transforms instead of Fourier transforms. It is important to notice that the coordinate dependence of the wave frequency has a profound effect on the nonlinear dynamics. The resonant wave interactions can now take place only over a limited range of wave trajectories which makes such interactions similar to the collision of discrete particles.

Similarly to the case of homogeneous turbulence considered in [13], the presence of a condensate changes the resonant wave

interactions from four-wave to three-wave if the condensate intensity exceeds that of the waves. A distinct feature of the inhomogeneous turbulence trapped by a potential is that if the three-wave regime is dominant in the center of the potential well, it is likely to be suddenly replaced by a four-wave dynamics when one moves out of the center beyond the condensate reflection points where the condensate intensity is decaying exponentially fast. Thus the same wavepacket can alternate between three-wave and four-wave interactions, with other wavepackets, as it travels back and forth between its reflection points in the potential well. (The wavepacket reflection points being further away from the center than the condensate's own reflection points).