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