Nonlinear waves and one-dimensional turbulence in nondispersive media

The main results of the theory based on the solution of Burgers' equation for large Reynolds numbers are reviewed. The basic properties of the arising stochastic regime, which is an example of strong turbulence, and its relation to hydrodynamic turbulence are discussed. Different stages in the evolution of a nonlinear wave are interpreted from the point of view of a flow of noninteracting particles. The statistical properties of Riemannian waves are analyzed for the stage of single-stream propagation. Methods for describing and the characteristics of the turbulence of sawtooth waves, forming at the many-stream stage, are examined. The self-preserving nature of this regime is demonstrated. The coupling of regular and random waves at different stages of propagation is examined. The possibility of describing the evolution of the average velocity with the help of turbulent viscosity is analyzed. Possible generalizations of the theory to related problems are discussed.