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We propose a new approach to study the long time dynamics of the wave kineticequation in the statistical description of acoustic turbulence. The approach isbased on rewriting the discrete version of the wave kinetic equation in theform of a chemical reaction network, then employing techniques used to studythe Global Attractor Conjecture to investigate the long time dynamics of thenewly obtained chemical system. We show that the solution of the chemicalsystem converges to an equilibrium exponentially in time. In addition, aresonance broadening modification of the acoustic wave kinetic equation is alsostudied with the same technique. For the near-resonance equation, if theresonance broadening frequency is larger than a threshold, the solution of thesystem goes to infinity as time evolves.

A reaction network approach to the theory of acoustic wave turbulence