In weak turbulence theory,the Kolmogorov-Zakharov spectra is a class of time-independent solutions to the kinetic wave equations. In this work we construct a new class of time-dependent solutions to those kinetic equations. These solutions exhibit the interesting property that ALL the energy is cascaded from small wave numbers to large wave numbers. The energy of the system is concentrated at{p=∞}and the energy function becomes a Dirac delta function at infinity.

The existence of this class of solution is a rigorous mathematical proof based on the kinetic description for the energy cascade phenomenon.