Coherent Structures and One-Dimensional Wave Turbulence
by
Frederic Dias
Ecole Normale Superieure de Cachan, France
Coauthors: Vladimir Zakharov, Russia

The problem of turbulence is one of the central problems in theoretical physics. While the theory of fully developed turbulence has been widely studied, the theory of weak wave turbulence has been less studied, partly because it developed later. Wave turbulence takes place in physical systems of nonlinear dispersive waves and it is not surprising that the theory of weak wave turbulence began to develop in connection with some problems of plasma physics as well as of wind waves. In most applications nonlinearity is small and dispersive wave interactions are weak. The weak turbulence theory is a method for a statistical description of weakly nonlinear interacting waves with stochastic phases.

The present talk deals with one-dimensional wave turbulence. Most of the talk is devoted to weak turbulence in selected wave equations. These equations are model equations, but consequences on physical systems such as ocean waves are discussed as well. The main conclusion is that the range in which the theory of pure weak turbulence is valid is narrow. In general, wave turbulence is not completely weak. Together with the weak turbulence component, it can include coherent structures, such as solitons, quasisolitons, collapses or broad collapses. As a result, weak and strong turbulence coexist. In situations where coherent structures cannot develop, weak turbulence dominates.

Date received: October 9, 2003