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 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(omega) at the zero frequency omega. 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 omega = 0 and a power-law asymptotic n(omega) -> omega(-x) at omega -> infinity x is an element of R+ . Finding it amounts to solving a non-linear 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 approximate to 4 . 7% which is realized for x* approximate to 1 . 22 . (C) 2021 Elsevier B.V. All rights reserved.