Self-similar evolution of Alfven wave turbulence

We study self-similar solutions of the kinetic equation for MHD wave turbulence derived in [1]. Motivated by finding the asymptotic behaviour of solutions for initial value problems, we formulate a nonlinear eigenvalue problem comprising in finding a number x ∗ such that the self-similar shape function f(η) would have a powerlaw asymptotic η−x∗ at low values of the self-similar variable η and would be the fastest decaying positive solution at η → ∞. We prove that the solution f(η) of this problem has a tail decaying as a power-law, and not exponentially or super-exponentially. We present a relationship between the power-law exponents in the regions η → 0 and η → ∞, and an integral relation for f(η) and x. We confirm these relationships by solving numerically the nonlinear eigenvalue problem, and find that x ∗ ≈ 3.80