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We carry out a systematic, direct numerical simulation (DNS) of the two-dimensional, Fourier-truncated, Gross-Pitaevskii equation to study the turbulent evolutions of its solutions for a variety of initial conditions. We find that the time evolution of this system can be classified into four regimes, which have qualitatively different statistical properties. In the first regime there are transients that depend on the initial conditions; in the second, power-law scaling regions, in the energy and the occupation-number spectra, appear and start to develop; the exponents of these power-laws and the extents of the scaling regions change with time and depended on the initial condition; in the third regime, the spectra drop rapidly for modes with wave numbers k > kc, and partial thermalization takes place for modes with k < kc; the self-truncation wave number kc(t) depends on the initial conditions and it grows either as a power of t or as log t; finally, in the fourth regime, complete thermalization is achieved and correlation functions and spectra are consistent with their nontrivial Berezinskii-Kosterlitz-Thouless forms, if we account for finite-size effect.
Author(s):
Rahul Pandit
Department of Physics, Indian Institute of Science, Bangalore
India
Vishwanath Shukla
Department of Physics, Indian Institute of Science, Bangalore
India
Marc Brachet
Laboratoire de Physique Statistique de l’Ecole Normale Supérieure, associé au CNRS et aux Universités Paris VI et VII, 24 Rue Lhomond, 75231 Paris
France