Statistics of nonlinear dispersive systems with small dissipation

We investigate the effect of a small dissipation on the turbulent motion in a nonlinear dispersive medium governed by the Kortewegde Vries (KdV) equation. A closed equation for the energy spectrum function is derived by making use of the derivative expansion method. From the solution of this equation we obtain the analytical expression for various statistical quantities such as the kinetic energy per unit mass, the energy spectrum function, the velocity correlation function and the 2nth order moment of the energy spectrum function. Asymptotic expressions of these quantities for large times are also given.