Exact scaling solution of the mode coupling equations for non-linear fluctuating hydrodynamics in one dimension

We obtain the exact solution of the one-loop mode-coupling equations for the dynamical structure function in the framework of non-linear fluctuating hydrodynamics in one space dimension for the strictly hyperbolic case where all characteristic velocities are different. All solutions are characterized by dynamical exponents which are Kepler ratios of consecutive Fibonacci numbers, which includes the golden mean as a limiting case. The scaling form of all higher Fibonacci modes are asymmetric L\'evy-distributions. Thus a hierarchy of new dynamical universality classes is established. We also compute the precise numerical value of the Pr\"ahofer-Spohn scaling constant to which scaling functions obtained from mode coupling theory are sensitive.