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 $n(\omega) \to \omega^{-x}$ at $\omega \to \infty \;\; x\in \mathbb{R}^+$. 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 $\approx 4.7 \%$ which is realized for $x^* \approx 1.22$.