Conclusion

In the present paper we have begun to develop a consistent statistical description of acoustic turbulence based both on the long-time asymptotic analyses (Section 3) and on the perturbation diagrammatic approach (Section 4). The first approach is more straightforward. The diagrammatic approach provides a systematic way of analyzing higher order terms in the perturbation theory.

Our main result is that the nonlinear corrections to the frequency is much smaller than the nonlinear damping of the waves. We find also the balance equation (4.43) which generalizes the simple kinetic equation for acoustic waves. One can show that the balance equation (4.43) has the same isotropic solution (Zakharov-Sagdeev spectrum) as the kinetic equation. However the kinetic equation for acoustic turbulence does not describe the angle evolution of turbulence: any arbitrary angle distribution is the solution of KE. In contrast, our balance equation (4.43) contains terms which describe an angular redistribution of the energy because of the non-zero value of the interaction cone, which is proportional to $\Gamma_{k12}$. But we have yet to show that this expression contains all such terms to this order.

One may imagine three very different ways of the angle evolution of anisotropic acoustic turbulence. The first one is a tendency to form very narrow beams with the characteristic width of about interaction angle. The second one is an approach to isotropy downstream to the large wave vectors. The last possibility is to form a beam with a characteristic width of about unity, exactly like it happens in the turbulence of waves with weak dispersion [17]. Another important question is: does the spectra of acoustic turbulence depend on the features of pumping or they are universal (independent of details of energy influx)? We intend to answer these questions (in the framework approximations we made in that paper) in our next project.

It is an exciting challenge to try to go beyond the approximations made here in order to understand whether the scaling index of the interaction vertex in the system of acoustic waves in two- and three-dimensional media must be renormalized or not. We acknowledge full spectrum of grants. V.L. acknowledges the hospitality of the Arizona Center for Mathematical Studies at the University of Arizona, where a portion of this manuscript was written.