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

Abstract 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 .