Self-similar evolution of Alfven wave turbulence

We study self-similar solutions of the kinetic equation for MHD wave turbulence derived in (Galtier S et al 2000 J. Plasma Phys. 63 44788). 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(eta) would have a power-law asymptotic eta(-x)* at low values of the self-similar variable eta and would be the fastest decaying positive solution at eta -> infinity . We prove that the solution $ \newcommand f(eta) 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 eta -> 0 and eta -> infinity , and an integral relation for f(eta) and x* . We confirm these relationships by solving numerically the nonlinear eigenvalue problem, and find that x* approximate to 3.80.