## Abstract

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 〈[Formula Presented]-[Formula Presented][Formula Presented]〉∝ln[Formula Presented] 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.

Original language | English |
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Pages (from-to) | 5095-5099 |

Number of pages | 5 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 54 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1996 |

Externally published | Yes |