# Chapter 12: On the energy cascade of acoustic wave turbulence: Beyond Kolmogorov-Zakharov solutions

In weak turbulence theory, the Kolmogorov-Zakharov spectra is a class of time-independent solutions to the kinetic wave equations. In this paper, we construct a new class of time-dependent solutions to those kinetic equations. These solutions exhibit the interesting property that the energy is cascaded from small wavenumbers to large wavenumbers. We can prove that starting with a regular initial condition whose energy at the infinity wave number $latex p=\infty$ is 0, as time evolves, the energy is gradually accumulated at $latex \{p=\infty\}$. Finally, all the energy of the system is concentrated at $latex \{p=\infty\}$ and the energy function becomes a Dirac function at infinity $latex E\delta\{p=\infty\}$, where E is the total energy. The existence of this class of solutions is, in some sense, a rigorous mathematical proof based on the kinetic description for the energy cascade phenomenon. We restrict our attention in this paper to the statistical description of acoustic waves.