Condensate turbulence in two dimensions

The nonlinear Schrödinger equation with repulsion (also called the Gross-Pitaevsky equation) is solved numerically with damping at small scales and pumping at intermediate scales and without any large-scale damping. Inverse cascade creating a wave condensate is studied. At moderate pumping, it is shown that the evolution comprises three stages: (i) short period (few nonlinear times) of setting the distribution of fluctuations with the flux of waves towards large scales, (ii) long intermediate period of self-saturated condensation with the rate of condensate growth being inversely proportional to the condensate amplitude, the number of waves growing as √t, the total energy linearly increasing with time and the level of over-condensate fluctuations going down as 1/√t, and (iii) final stage with a constant level of over-condensate fluctuations and with the condensate linearly growing with time. Most of the waves are in the condensate. The flatness initially increases and then goes down as the over-condensate fluctuations are suppressed. At the final stage, the second structure function 〈|${\mathrm{\psi }}_1$-${\mathrm{\psi }}_2$ ${\mathrm{|}}^2$〉∝ln${\mathit{r}}_{12}$ while the fourth and sixth functions are close to their Gaussian values. Spontaneous symmetry breaking is observed: turbulence is much more anisotropic at large scales than at pumping scales. Another scenario may take place for a very strong pumping: the condensate contains 25–30 % of the total number of waves, the harmonics with small wave numbers grow as well. © 1996 The American Physical Society.