Numerical analysis of a self-similar turbulent flow in Bose–Einstein condensates

We study a self-similar solution of the kinetic equation describing weak wave turbulence in Bose–Einstein condensates. This presumably corresponds to an asymptotic behavior spectrum evolving from broad class initial data, and it features non-equilibrium finite-time condensation n(ω) at zero frequency ω. The is second kind, satisfies boundary conditions corresponding nonzero constant (with all its derivative being zero) ω=0 power-law n(ω)→ω−x ω→∞x∈R+. Finding amounts solving nonlinear eigenvalue problem, i.e. finding value x* exponent x for which these two can be satisfied simultaneously. To solve this problem we develop new high-precision algorithm based on Chebyshev approximations double exponential formulas evaluating collision integral, as well iterative techniques integro-differential shape function. procedures allow achieve with accuracy ≈4.7% realized x*≈1.22.