Theory of Laminated Turbulence: Open Questions

The roots of the theory of nonlinear dispersive waves date back to hydrodynamics of the 19th century. It was observed, both experimentally and theoretically, that, under certain circumstances, the dissipative effects in nonlinear waves become less important then the dispersive ones. In this way, balance between nonlinearity and dispersion gives rise to formation of stable patterns (solitons, cnoidal waves, etc.). Driven by applications in plasma physics, these phenomena were widely studied, both analytically and numerically, starting from the middle of the 20th century. The main mathematical break-through in the theory of nonlinear evolutionary PDEs was the discovery of the phenomenon of their integrability that became the starting point of the modern theory of integrable systems with Korteweg-de Vries equation being the first instance in which integrability appeared. But most evolutionary PDEs are not integrable, of course. As a powerful tool for numerical simulations, the method of kinetic equation has been developed in 1960-th and applied to many different types of dispersive evolutionary PDEs. The wave kinetic equation is approximately equivalent to the initial nonlinear PDE: it is an averaged equation imposed on a certain set of correlation functions and it is in fact one limiting case of the quantum Bose-Einstein equation while the Boltzman kinetic equation is its other limit. Some statistical assumptions have been used in order to obtain kinetic equations; the limit of their applicability then is a very complicated problem which should be solved separately for each specific equation.